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r / LOGARITHMOLOGIAr^7 

ORTHEWHOLE , M 3> 8 

i DO C T R I N E, 

[> xo Car I t h m s^ 



ft. 



Common and hogiftical^ 

IN 

THE.ORY and PRACTICE. 

IN THREE PARTS. 
Part. I. The THEORY of Logarithms; 

Shewing their Nature, Origin, Cotfftru^lion, and Properties, 
demonftrated in varioas Methods, was. i. By Plain Arithme- 
tic. 2. By the Logarithmic Carve. 3. By Dr. Halley's 
Infinite Scries. 4. By Fluxions, c. By ^c Properties of the 
Hyperbola. 6. fey the Equiangular Spirnl. 7. By a Loga- 
rithmic infpeftional Scale of twenty- two Inches length. With 
the Conftru^ion of the artificial Lines of Numbers, Sines, 
and Tangents. Alfo the Nature and Conftrudlion of Logiftical 
Logarithms. The whole illutlrated and made eafy by many 
<and fuitable ExaB^ples. 

Part II. The PRAXIS of Logarithms; 

Wherein all the Rules and Operations of Logarithmical Arith- 
metic, both Common and Logittical, by Numbers and Inftru- 
ments, are copioufly exemplified. Together with the Ap- 
plication thereof to the feveral Branches of Mathematical 
Learning. 

PartIIL a Three-fold canon of Logarithms; 

In a new and more compendious Method than any cxtint ; 

r I. A Canon of Logarithms of Natural Numbers. 

* Vi%. X 2. A Canon of Logarithms of S 1 n fis and Tangents. 

\ ^3. A Table of Logistical Logarithms. 

The whole being a CompUat Syftem of this mod ufeful Art ; 
and enrich'd with all the Improvements therein from its Ori- 
ginal to the Prefcnt Time. 

By B E N 7 A M I 1^ MARTIN, 

Author of the Philological Library of Literary Arts and 
Sciences^ &c. 

LONDON: 

Print^ for J. Hodges, at the hooking-Qlafs on London-Bridge* 

M.pcg.xxxx. 



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^ THE 

1 



iP R E F A G E. 



TBO Logarithms fnay juftfy be efteenid 
the principal Invention ^modern Ages, 
on account of their excellent and tnojl 

\[ fcxtcniivc Ufe in Mathematical Literature, yet 

it may ^ith equal Truth be faid^ that little 
inoh is known of them^ generally Jpeaking^ 
than theif praftical Ufe infome Rules ^/com- 
mon Arithmetic and trigonometrical Calcula- 
tions i {and how few are perfe£i in this !) 
For (faith the great Improver of this Art^ the 

^ learned T)r. Halley) " / fivU very few of 

*^ thofe^ whomakeconftantufeofLo^^mhms^ 
" have attain d an adequate Notion of them i 
^^ to know how to make or examine them^ of 
" to underfland the Extent ^/ the \5i^ of 
" them : contenting themfelves with the Ta- 
*' bles of them, as they find them, without 
^^ daring to queftion them, or caring to know 
'^ how to redify themy Jbould they be found 
" amifs i beingy I fuppofe,^ under the Appre- 
'^ herfion of fame great ^:jficultjf therein, &c/' 

A 2 ^ For 



iv PREFACE. 

For the fake of fuch Terfons the following 
Trcatife is principally intended i wherein they 
fwill find every thing necejfary relating Jo the 
Yheory and VxoiQticc of this admirable Art. 

For, as to the Theory, {the principal Part, 
and fo very rarely known) I have exhibited 
all the Methcxls, whereby it has been taught 
and explain d by the Inventor, and fever al 
Improvers thereof fince his time y as by the 
Extraftion of Roots i by the Logarithmic Curve 
from jDr. Kcil > by an Infinite Scnts^from T>r. 
Halky i by the Method of Iluxions, from Mr. 
Ditton i by the Hyperbola, /r^w? Mr. Domky i 
by the Equiangular Spiral, from Halley, Wallis, 
h'C. by the /^rg*^ Logarithmic Scale, of my awn 
conftruding : the like to which, for Large- 
nefs, was never before publijhed j for a com- 
pleat Account of which fee Chap. X. of the 
Theory. 

I fay, by all thefe various Methods / have 
endeavour d to explain, illuftrate, and facili- 
tate the Knowledge of the Nature, Proper- 
tics, and Conftrudion of thofe excellent Num- 
bers, called Logarithms. / have alfo exem- 
plified the Manner of making Logarithms yj?r 
the prime Numbers, by many and different 
Examples, afid in feveral ways s and have 
taken all poffible Care to render this mojl ab- 
ftrufe a7id difficult ^art, as eafy and intelli- • 
gibic, as the Nature of the SubjeSt will 
admit. 

Having 



PREFACE. V 

Having thus explain d the Nature of Lo- 
garithms, / then fiew how they are laid wn 
Inftruments, and thereby the Cooftruftion of 
the Artificial Lines of Numbers, Sines, and 
Tangents i firji contrived by the famous Geo- 
meter y Mr. Gunter of Grefliam-Collegc s an4 
for that reafon they are ftill called Guntcr's 
Line, and all together Garner's Scale. 

Lajilyy I have largely explain' d the N^ 
ture^ and Jhewn the Manner of making or 
conJiru£ting the tx)giftical Logarithms, accor- 
ding to ShakerleyV and Stictt's Form thererf^ 
and which I have notfeen done by any otier 
Hand. 

Thefe things together willy I hopCy be al^ 
low'd to make a regular, univerfal, and com- 
pleat Theory of Logarithms, common and lo- 
giftical, for Integers and FraiiionSy numeri- 
cal and inftrumental ; and fuchy as for Bre- 
vity, yet Copioufnefs and Variety, has not 
been before extant. 

As touching the Praxis, or Ufe ^Loga- 
rithms, which makes the fecond Tart of this 
PForky I have made it as compleat and per- 
fe£t as poffible^ having illuftrated all the 
Rules of Praftice with all theV^ncxy ofUx- 
ampJcs / could devifey that were necejfary. 
■^nd that none may be unapprised of the mofi 
^xtcxifivc Service of Logarithms in the Mathe- 
^^^tical Difdplines, I have applied them to the 

Arithmetic 



in t^REFACE. 

Arithmetic of all kinds of Number i ^ to Tri" 
gonomctry, in the Solution df all Cafes of 
Plain and Spherical Triangles i to Mcrc^tors 
Sailing particularly y jhowinghow all its Cafes 
may be refolvd folely by the Canon of Loga- 
rithmic Tangents ; to the Mcnfuration 4if Su- 
perficies i^«/Solids^ ^c: All which are fun- 
damental Operations, and may each of them 
be extended or branched out into particular 
Sciences i but that would have been too te- 
dious a Taskj and not ahfolutely necejfary to 
^ ^ejign. I have there fare only applied the 
Doftrine of Mcnluration to the Arts of Gzn^- 
ing, Timber-Meafure, ^«// Surveying i becauje 
they are the mo(l common and n^ceffary Arts 
in Life s and beeaufe the Ufe of Gunter'S 
Scale, and Slidiag-Rule {though before fully 
taught^ and all along applied i yet) in theni 
is more extenfive and various than in any o- 
ther Arts: and therefore I have taken care 
not only tojkew all the different ways ofufing 
thofe inftruments, but likewife the Rationale: 
of every Operation ^ a Matter of the greatefl 
Importance^ and too often negle^ed^ in Books 
which treat thereof 

Laftly, I have in the laft Chapter given a 
Variety of Examples of the life of Logiftical 
IvOgarithms in ?^^ practical Parts i?/*Aftronomy; 
4)oih with refpeB to Time and Motion, have 
made it appear that Street's Logiftical Loga- 
rithms arifwer all the Ends of Slaakerlcy j i 
^nd how they are * to be ufed along with the 
Common Logarithms of Numbers, Sines, and 

Tangents; 



PREFACE. vii 

Tangents. And throughout this fecoAd Part, 
as w^ll as tj^e firft, you will fnd 4 great 
Variety of new and ufeful Particulars^ not 
here tfi be exprefsd. ♦ 

The Third Part of this Work canfijls ^f a 
xjtixt^'iold Canon of Logarithms, viz. (i.) Of 
fomnum Numbers from i to loooo/ oAd is 
Gifficient for any Number under 1 0000000^ 
proper Rules. (2). Of Sines 49?^ Tangents /v 
every Degree and Minute of the Quadrant. 
(i-) ^f L^i^i^'i Logarithms of Mr. StrcctV 
Form. Concerning which Tables y Ifballcnly 
Qbfsrve two things in general^ viz, 

Firfly that they are here contrived in a 
new and moji compendious Form, equally cafy 
and ufcful as thofe of the common Formi thd 
in this they take up but one half the room^ as 
they do in that. An Abbreviation very conh 
modiouSy and I hope will prove acceptable. 

Secondly ^ tiie Cwrte^nefs of thefe is a 
matter of the tafl Concerny and the great eft 
Argument to recommend them. In order to 
prove thiSy J need only fay^ that thofe large 
Tables of Mr. Sherwin j ape granted to be the 
moft correSi of any extant y from the maft 
(^tQfal^and exquiiite Method he took to make 
ihemfoy which fee in his Preface. 

From thefe large Tables thus correify I 
have made minCy every Figure of the two 
^rft Tables with my own Hands in doing 

which 



vifi PREFACE. 

whichy I difcoverd fever al Error s, here and 
there y as I went along, in them^ as exuEt as 
they were, which accordingly correcting in 
mine, lean, I prefuthe, juftly pronounce my 
Tables thi mofi certain and exaft, as well as 
the moft compendious of any in being. 

Having thus largely declared the fever al 
Tarts of this Work, and Jhewn the Reader 
whdt an ufeful Variety he may expert to meet 
with both in the Theory, Praxis, and Tables, 
of this moft 4ifeful and excellent Art } I muft 
leave it to himfelf to ufe or reject it, as he 
(ball judge of the Merits thereof 




THE 



(«) 




THE 

TABLE 6f CONTENTS. 

PART L 

CHAP. I. 

Op /i6^ Definition, Origin, and Natvvle 
^Logarithms. p. i 

CHAP. IL 

Of the Method cf making a Table of Loga- 
rithms i^ Plain Arithmetic. 9 

C H A P. III. 

T'be Doctrine ofibe Nature and Prope rties 
of luOGAK I THUS farther explained and illuftratcd 
by means of tbe Logarithmic Curve. 18 

C H A P.^ IV. 

2*i&^NATURE (?/■ Logarithms and their IitvtCES^ 
when tbe Numbers are Fractions, farther ex^ 
plained by tbe hoGAKiTHMic Curve. 25 

C H A P. V. 

fTbe Original Conftruftion of Logarithms by tbe 
Lord Ne per, and tbe Change thereof to tbe pre- 
fent Form by Himfelf tf»i Mr. Briggs, explained 
and illuftratcd by tbe Looa rithmicCwrve. 35 

CHAP. VL 

yf Method of Conftrufting the Logarithms, de- 

, rived and demonfirated from tbe Nature of Num- 

^ B^RS only^ by Dr. Edmund Halle y. 40 

a CHAP. 



X CONTENTS,,. 

CHAP. VII. 

The JLoc arithmicSeries aforegoing demonfirated 
alfo by Fluxjons^; and from the Nature of the 
HypERpoLA. * p. 50 

CHAP. VIII. 

?J^ Method of Conftruifing Log a Kir hub by the 
Infjnite Series, exemplified and illufirated. 53 

C H A P. IX. 

Qf /^^* Logarithmic Spiral ; and the Nature 

and Qor[&xw&\on of a Table of Meridional 

Parts or Nauticfl Meridjan Line, deduced 

therefrom. 64 

CHAP. X. 

Of the Conftruiftion of a Large Logarithmic 

Sc ALE, exhibiting^ by Infpeftion, a diftind Idea 

^/i>^ Nature ^» J Agreement of Numbers and 

their LoQARiTHMS. 71 

CHAP. XI. 

Of the Conftnwftion of the Artificial Lines ^N^m- 
b|:rs, SiNfs, ^;/i Tangents, ^ i»^tf»i ^/Lor 

GARITHMS. '73 

CHAP. XII. 

Of the Manner of ufing the Tables of Log a-? 
' RiTHMs/» Practice » and of the Pre requi- 
sites /i^^r^/^. 835 

CHAP. ?crii 

Of the Origin and Conftrudion of Shake rleyV 
>?»^St;ieetVLogisticalI^ooarithm3. 93 

!> A R T IL 

G H A P. I. 

<y /i&? Rules u/ Addition, SuBSTRACTioii,Mut,"r 
. TjPLicATjoN, ^nd pivisioN of the Indices ^» 

LoGARItHMS, * 103 

C H A F. 



r 



CONTENTS. si 

CHAP. II. 

f^MutTIPLICATlON tf)iiDlT|SION ^ Wh6Ls 
NuMBEI^S^»^DECIM4I-SbyL0OARITHMS.I0$ 

CHAP. III. 

Of Raifing Powers andtU £xt b act ion ^Rqot$ 

>y LOGARITHMS. P-II5 

CHAP. IV. 

Ofthi various Ru lbs ^t/Troport k>k, tfni of findings 

M£A9PROPORTIONA|.si^LOGARITHM8.I2C 

CHAP. V. 

ISiMPLB Interest by Logarithms. 128 

CHAP. VL 
Compound Interest by Logarithms. 138 

CHAP. VIL 
ViTLG A R Fractions i^ Logarithms 148 

CHAP. VIII. 
*Puo&£crMAL Akithu^t ic pirfotn^d iy Looa« 

RITHMS. 152 

CHAP. IX. 

flT^^OpERATioN ^/itf common Rules ^ Arith- 
metic ty Inst RujfE NTS j viz. the Loga- 
rithmic Scale, tf»rfGuNTER's Line mtb 
/i&^ Compasses, ^»^^ir/i&^SLiDiNG-RuLE. 158 
C H A P. X. 

^keiJbeth the Analogies for the Solution of all 
the Cases of Plain and Spherical Tri an-p 
G L E s, both Right and Oblique angled. 1 67 

CHAR XI. 

y'ifrtf Solution of Plain TRiANGLE&iytbeCA^ 

!f O N (?/ Lo q 4 R I T H M I C S I N E S ^l»^ T A N G ¥ N T S ; 
^GuNTEB*sScALE^WCOMPASSES,tf»^^JF/i&^ 

l idinc-Rule. 173 

C H A p. XII. 

^e$OLvxiQN of Spherical Triangles ly 

Logarithms, iy Gunter's Scai;-e, and by 

/i>^ Sliding-Rule. 1*8$ 

" ' ^ H A P. 



I 

301 CONTENTS. 

CHAP. XIII. : 

M?RC A tor's Sailing performed ly ib^ QakoiI 
£/■ Logarithmic Tangents^ without the 
Meridional Parts. p. 2ocr 

CHAP.. Xi V. 
Of the Mensuration (j/'SuperfIcies and So- 
L I D Bo D I E s ^j^ Logarithms. 207 

C H A P. XV. . 
^eDo&rine ^Mensuration applied to Gaug- 
ing, Meafuring Timber, and Surveying; 
wherein /i&^ Praftical Ufe of. theVhAiti Scale» * 
and Sliding Ru l e for tbefe Purpofes^ is clearly 
^xplain^d. 222 

CHAP. XVI. 
^n>e Prafticaf Usfi of the Logistical Loga- 
rithms. 239 



-ii^ 



f. PART ill. 

yf Canon yLoGARiTHMs/?r Natural i^uM^ 
BERs, /||ij» I /^ loooo. 249 

A CanonT?/' Logarithmic Sines ind Tan- 
gents. ^ 

A Canon or Table of Logistical Loga- 
rithms. 



L O G A- 



] 














i 




LOGARITHMOLOGIA. 

PART I. 

The 7%eory of Log AKiTHMs* 



C H A P. I. 

Of the Definitioriy Origin^ and Natun of 
Logarithms. • 



I "T 



^HE bcft Definitim of thofe Numbem 
we caii Logarithms^ is containM in thb 
very Name or JVerd {Logariibm) it* 
felf 5 for it is compofed of die two 
Greek Word« xiym «piO|uo\, which properly or lite- 
rally %nify, a Number of Rations: and a Logarithm 
is no other than a Number, which denotes or fliews 
what Number of Rations is contained between Unity^ 
and fhat^ Number of which it is ikid to be the Lo^ 
garitbm. 

2. Whence 'tis evident^ that In order to have a 
clear Notion of Logarithms^ 'tis abfolutelv neceffiuy 
to underftand firft, and that very well, what is 
meant by the Word Ratic^ ot R^atio^s^ as here ufed 
in the Definition of Logarithms, and making an ef« 
fcntial part thereof. 

3. Ratioy then^ is a certain mutual Habitude of 
Magnitude^ of the fame kind^ according to ^antity. 
This is Euclid^s Definition : in which four things 
muft be obferv'd j as (i.) he {kith Ratio is a certain 
mutuiU Habitude} by which h^ pieana no more than 

B ' what 



a Of the Definition, Origin^ and 

whaCt we commonly call xht Proportion of any two 
things of a like fort tojacb other ^ when by ys they* 
arc compared together. (2.) He ufeth the general 
Word Magnitude to denote, that all SubjeSfs of ^an- 
tity^ 9S Numbers^ Lines^ Superfictes, and Solids^ are 
capable of fuch Ratioj Habitude^ or Proportion^ as 
aforefaid. (3.) He adds this Reftridion, of the fame 
kind 5 thereby infinuating there can be no Ratio or 
Proportion of ^antity between Magnitudes ctf*a dif- 
ferent kind ; thus we cannot compare a Line to a 
Superficies ; becaufe the Quantity of a Line is cfti- 
mated in Length only, but the ^antity of a Super- 
ficies arifeth from the Joint Confideration of Length 
and Breadth, or the Produft of each, and fo im* 
porteth Space ; which is entirely different from a 
Line^ and therefore thefe two things cannot be the 
Terms of Ratio or Comparifon. (4.) Laftly, he 
fays, this Ratio is according to ^antity ; that is, 
we compare Magnitudes in this Cafe, only to ob- 
lerve and maintain the Proportion of Greatnefs^ or 
Bulk which is between them ; or to find how often^ 
or how many times^ one leffer Magnitude is contained 
in another greater Magnitude \ negleAing all other 
Confiderations and AfFedions of the faid Magni- 
tudes. 

. 4. Having thus confider'd the general Nature of 
Rations or Proportions ; I fhall apply it to Numbers^ 
as they are immediately the Subject of Logarithms. 
The Ratio therefore, or Proportion of a Number to 
a Number is two-fold ; for firft the Ratio of a greater 
Number to a leffer may confifl in tht Addition of 
fome certain Number to that lefjer Number '^ thus the 
Ratio of 6 to 2 is made by adding 4 to 2. And 
if from Unity you begin the conftant Addition of 
the fame Number, fuppofe 2, you will then have a 
3eries of Numbers, whofe Differences will be the 
feme, as i, 2, 4, 6, 8, 10, 12, i^c, and fuch Num- 
bers are (aid to be in Arithmetical Proportion or. Pro- 

• . grejftoni 



Chap. I. feature g/*LoGARiTH M5. 3 

• gr^ffion 5 and this common Differenced the Terms, 
as here 2 is called the Ratio of the Progreffion. 

5. Secondly, the Ratio of a greater Number to a 
lejfer may'confift in a Multiplication o( the Lejfer 
Number by fome other Number 5 thus the Ratio of 
12 to 4, is made by multiplying 4 by 3 ; and if 
from Unity you begin a conftant Multiplication by 
th^ fame Number^ fuppofe 2, you will then have a 
Series of Numbers, as 1, 2, 4, 8, 16, 32, 64, £s?r, 
which are faid to be in Geometrical Proportion^ or 
Progrejfton ; and the common Multiplier^ as here 2, 
is called the Geometrical Ratio of this Progreffion. 

6. Wherefore in the two Series or Progreffions of 
Numbers, viz. 

Arith. I, 2, 4, 6, 8, 10, 12, 14, (Sc. 
Geom. I, 2, 4, 8, 16, 32, 64, 128, 6?^. 
*tis eafy to obferve, that as the /econd Term ex- 
ceeds the firft by one Ratio, fo the third Term ex- 
ceeds the firft by two Ratio's, the fourth by three 
Ratio's, the fifth by four Ratio's; fe? f . Thus in the 
Arithmetical Series, the Ratio of 4 to i, is dottle 
of the Ratio of 2 to i ; the Ratio of 6 to i, is /r/- 
ple the Ratio of 2 to i ; the Ratio of 8 to i, is 
quadruple the Ratio of 2 to i, fcfr. And in the 
Geometrical ISeries, the Ratio of 4 to i is the du^. 
plicate of the Ratio of 2 to i ; the Ratio of 8 to i 
is triplicate of the Ratio of 2 to i, and the Ratio 
of 16 to I is quadruplicate of the Ratio of 2 to i, 
and fo on. Where 'tis to be obferved, that tho 
Words, double^ triple^ quadruple^ &c. are proper to 
the Ratio's of the Arithmetical Series^ and imply the 
Addition of them only ; but the Terms duplicate^ . 
triplicate^ quadruplicate^ Stc. are proper to the Geo-, 
metrical Series^ and imply ,ithe Multiplication of thofc 
Ratio's. 

7. What has been thus far related of the Doftrine 
of Ratio's, is fufiicient for our prefcnt purpofe,. vi%. 
the underftanding the Nature of logarithms.: For 

B 2 fuppofe 



4 Of the Definition^ Origin^ and 

fuppofe a Series of Numbers in Arithmetical Progref" 
fton^ beginning frpm o, and whofe Ratio^ or com- 
iiion Difference > is Unity or i ; and to them be 
adapted a Series in Geometrical Progrejfion^ beginning 
from Unity ; and whofe common Ratio be any af- 
fign'd Number, fuppofe 2, as before ; then will 
thofe two Series ftand as below : 

. iArith.o. 1.2.3. 4- 5- 6. 7, 8. 9 &?f. 
• *Geom;i. 2.4.8. 16.52. 64.128. 256.512.6?^. 

8. 'Tis evident the Numbers in the firft Series 
fliew the Number of Ratio's between their corre- 
fpondent Numbers and Unity in the fecond Series. 
For inftance, the Figure 2 in the firft Series, fliews 
the Ratio's hptween its correfponding Number 4 and 
I, in the lower Series, are 2 ; the Numbers 5, 7, 9, 
in- the upper Series, fliew the Number of Ratio's be- 
tw(*eQ their correfponding Numbers 32, 128, 512, 
and I or Unity, in the lower Series, to be refpeftively 
5, 7, and 9 ; or that the Ratio is fo often repeated 
from Unity to thofe Numbers, and confequently fo 
often compounded in them ; or farther, that the 
Ratio of 32 to I is compounded of five times the Ra* 
tio of 2 to I ; and the Ratio of 128 to i, of feven 
times the Ratio of 2 to i ; and the Ratio of 512 
to I, of nines times the Ratio of 2 to i. 

9^ Wherefore fince the Numbers in the uj^^ Se- 
ries (Hew the Number of Ratio's contain'd between 
their correfponding Numbers and Unity in the lower 
Series ; therefore thofe Numbers in the upper Series 
in Arithmetical Ptogreflion fliall be the Lagarittms 
of the Numbers in the lower Series of Geometrical 
Progreffion, and that according to the Definition of 
J^ogarithms beforegoing. 

iQ. From the &id Series, 'tis fiuther manifeft, 
that the Froduft of any two Terms in the lower 
Series corrcfponds to the Sum of their rcfpeftive 
Terms in the upper. Series. See the folbwing £x* 
ampler 

^ Arith. 



Chap. I. iVij/«r^ ^Logarithms. 5 

Logar. {2 + 3= 5i2+ 4== 6^3+ 6« 9. 

N:Nl;m'{^''^ = 3254x i6«64-, 8 x 64=512. 

Alfo if any two Numbers in the lower Series be di- 
vided the one by the other, the Quotient thence a- 
rifing (hall correspond to the Difference of the respec- 
tive Numbers in the upper Series. Examples in 

Arith. 5— 3 = 2j 6 — 2= 4; 9—3==: 6. 
Geom. 324- 8s=s4i 64-^4= 165 512 -7- 8«8 64« 

1 1. And univerfally, if any four contiguous Num* 
bers be taken in the fecond Series, as the ProduA 
of the Extremes is equal to the ProduA of the 
Means ; fo in the firft Series of the four correfpon* 
ding Numbers, the Sum of the Extremes will be 
equal to the Sum of the Means; as in the Examples 
bdow. 

Arith, 1+ 4=2+3= 5; 3+ 6=s 4+ 5=5 ^. 
Geom. 2 X 16=4 x 8=32 •, 8 x 64=16 x 32=512. 

and vice i^erfa. 

12. Again the Square^ Cube^ (^c. of any Number 
in the lower Series of Geometricals will be anfwer*d 
by doubkj triple^ &c. the correfponding Number 
in the upper Series in Arithmetical Ratio ; for Ex- 
ample ; 

Square j^"*- ^'^"^ f* ^ 3= «><^^^=^- 
^ *Geom, 4x4=i6i 16x16=256-, IX isai. 

The Cube \ Arithmet. i x 3=3 ; 3x3= 9. 
■'^^'""^tGeomet. 2x2x2=8. 8x8x8 = 512. 

And the Converfe of this Article is alfo true ; as is 
evident enough without example. 
: Jfc^i fr appears then by thele fix laft Articles, that 
jL^arifims are a Series of Numbers in Arithmetical 
Pfagreffion^ fo fitted and adapted to another Series of 
Numbers in Geometrical Progrejfion^ as that each 

I Term 



6 Of the Definition y Origin ^ and 

Term of the firft Jhall expound (or be the Exponent 
of) the Ratio of its correfpondent Term to Unity in 
the fecond Series. And that on this very Principle : 
For every Addition^ Subjiraciion^ Multiplication^ or 
Divijion of the Logarithmtc Numbers there corre- 
fponds a mutual Multiplication^ Divifion^ Involution^ 
and Exfrailion of the refpeSlive Terms in the Geo- 
metrical Series. 

14. Now 'tis a Matter entirely arbitrary or indif- 
ferent, what Number be made tht firft Term in ei-. 
ther Series; for fince tht firft are made by equal 
Additions^ the latter by equal Multiplication^ be the 
Ratio what it will, th^ former will ftill be the Loga- 
ritbms of the latter ; as is evident in the Table ad- 
joined. 





Series of Logarithms. 


6 


I 


. ' I 


3. 


6 


7 





18 


2 


2 


5 


7 


12 I 


10 


54 


4 


3 


7 


8 


17 2 


20 


162 


8 


4 


9 


9 


22 3 


30 


486 


16 


5 


II 


10 


27 4 


40 


1458 


32 


6 


13 


II 


32 5 


50 


4374 


64 


7 


15 


12 


37 6- 


60 


1 3 122 


128 


8 


17 


13 


42 7 


70 


39366 


256 


; 9 


19 


14 


47 8 


80 


I I 8098 


512 


10 


21 


15 


52 9 


90 


354294 


1024 


II 


23 


16 


57 10 


100 



15. Wherefore to tht fame Series of Proportion 
nals there may be an infinite' Number of 5m^j or 
5rtf/^J of Logarithms contrived ; and vice verfa. Yet • 
of tf//thofe infinite kinds of Logarithms, only thofe. 
whofe firft Term is o, and the common Difference i, 
10, 100, 6fr. are adapted for ule. Becaufe if the 
firft Term be a fignificant Figure, we muft necefla- 
rily have refpcdl: to it in ufe 5 and fo, in this cafe, 

four 



Chap. I. Nature of Log AKiTU his. 7 

four Terms of the Losaritbmical Series becomes un» 
avoidable •, whereas if o be the firft Term, throe 
other Terms only fuffice in the Multiplication of any 
two Proportionals whatever: for the Sum of their 
Logarithms will point out the ProduSl^ and /hew 
its Place^ that is, its Difiance from, or Ratiif to, 
the firft Term of that Scale of Proportionals. But 
if the firft Term in the ScaljB of Logarithms were 
fignificant, it muft be fubdufted from the Sum of 
the Mean, in order to find the Produdl of the two 
Proportionals^ 6?r. as before: and fo we (hould per- 
petually have double Labour in every Operation. AH 
this is evident from the difiFerent Series of Loga- 
rithms in the foregoing Table. 

16. This being the Nature, and fiich the won- 
derful Properties of thofe Numbers called Loga- 
rithms, 'tis natural to fuppofe that he who firft dif- 
cover'd them, would make fuch a noble Difcovery 
as fubfcrvient as poffible to the Ufes of Life, for the 
general Benefit of Mankind, but more immediately of 
Artifts or Mathematicians. 

17. Now to do this, 'twas 
necefiary to calculate and fit 
a Scale or Canon of Loga- 
rithms to all Numbers which 
Men commonly make ufe of 
in Bufinefs, that fo the Fa- 
tigue and Labour of Multi- 
plying^ Dividing^ &c. large 
Sums or Numbers might be 
avoided by only the Addition or Subftradlon, 6? r, 
of Logarithms. And of confequence nothing lefs 
than a general Table of Logarithms for all Numbers 
betWeen Unity, or i and 1 000000 or 1 0000000, could 
fuffice ; becaufe, no Number muft here be wanting, 
as in the other intermitting Series i, 2,4, 8, 16, £5?^, 
fince general Ufe requires them all, and therefore 
their Logarithms. 

18. 



Proport. 1 Logarith. | 


I 


0.0000000 


10 


1. 0000000 


100 


2.0000000 


1000 


3.0000000 


1 0000 


4.0000000 


I 00000 


5.0000000 


loooodo 


6.0000000 



8 Of the Definition^ Origin y &c. . 

1 8. Towards this 'twas eafy to find Logarithms 
for a Series of Numbers proceeding in a Decuple 
Propmitm or i?^//^ from Unity, as i, lo, lOO, 1060, 
6fr, in the Table above. For putting o r= Loga- 
rithm of I 5 and I = Logarithm of ip ; the Loga- 
rithm of 100 will be 2 ; and of 1000, 3 ; and fo on 
from what I have already faid. 

19. But the Difficulty dl confifted in finding the 
Logarithms for the intermediate Numbers between 
I and 10, to and 100, 100 and 1000, tfc. or thofe 
of them which are called Prime Numbers^ becaufe 
they being once found, the Logarithms of all others 
are cafily obtained, as we Ihall fee by and by. 

20. Now tho* the Numbers i, 2, 3, 4, 5, 6, 7, 
8, 9, are not in Geometrical Proportion (but indeed 
in an Arithmetical one) and fo their Logarithms not 
to be obtained like the Logarithms of thofe which 
are fo, yet as the Ratio of 10 to i, (hews only what 
Diftance^ or bow far 10 is from Unity in the Scale 
of Proportion, and that the Numbers 2, 3, 4, 5, 
•6?f. poflefs feverally a certain part of that Diftance, 
if the iaid Ratio or Diftance of lo to i, be fuppofed 
;to he divided into a vaft number of equal Parts^ 
fiippoie looooooo, ^c. 'tis evident a certain number 
lof thofe equal Parts are to be allotted to the Num- 
bers 2, g^ 4, 5, 6?r. which (hall as truly exprefs the 
J}^ja$ces from, or the Rations of thofe feveral Num- 
bws to Unity, as that of 10 to Unity is exprefs'd by 
^ wiiole Number 1 0000000. 

^i. This indefinite Number (loooooo, (Sc) of 
ijpial Part 5^ into which the whole Ratio of 10 to i, 
fe divided, may be conceived as a Number of fo 
tnany finall Rations or RatiuncuU^ fince they are all 
equal to each other. And fo fince the whole Num- 
ber of RatiuncuU between 10 and i, is the Loga- 
rithm of I etc I ; therefore that Number of thofe 
RatiuncuU which lie between, or exprefs the Di- 
flftnceof 2, 3, ^c. to i, fhall alfo be as much the 

Logarithms 



Chap.tl. OfmaiingaTaMeePLoQAiLiTHMt.^ 
Li^arisbm of thofe NumbQrs 2» j, 6fr. or, ntfacr, 
of their Ratio's to Umty. Atid thus it appears that 
even the Logarithms of 2, 3, 4, .5, fc?c. ^and for 
the fame Reafon) 11, I2, i j, i^c. 21, 22, 23, £«?f, 
loj, 102, 103, i^c. may be found to as gteat Ex- 
adtnefs as is ncceflary. The Method of doing this 
by Numbers, ihall be the Subjeift of the next Chap- 
ten 

C H A P. U. 

Of the Met hod of making a Table of Logarithms 
by plain Arithmetic. 

I. T TAving explain'd the Nature and Properties 
X i of Logarithms, and fhewn not only how 
they are adapted to Series of Geometrical Pro- 
portionals, but alfo by what means they are to be 
calculated for all Numbers from Unity to any large 
Number either above or below it : 

2. Ths, Method or Procefs it felf, then, of doing 
this, is the next thing to be propofed ; an ar- 
duous 7ajk this, to him that firft aggre{s*d it ! At 
fuchan Enterprize, he indeed might truly have laid. 
Hie Labor, hoc Opus eji. And this any but an i»- 
human Reader will be convinced of, by viewing the ' 
following tedious and laborious Procefs for gaming 
the Logarithm of one fmgle Prime Number, and 
that only to feven Places of Figures. 

3. For, as I faid before, 'tis fufikient to produce ' 
the Logarithms of Prime Numbers only , becaufe, by 
the Additioft, Subftradtion, 6fr. of thefe, we obtain 
the Logarithms of all Compofit Numbers, with 
Eafe ; as will appear further on. Now, tho* there 
are but three Prime Numbers, viz. 3, ^, 7, between 

C I 



'io Of the Method oj making 

I and 10 ; yet» becailfe the Logarithm of no Num* 
ber between i and lO can be round by the Loga- 
rithms of any of the Proportionals I, 10, IOO9 IOOO9 
(Sc. it follows^ that the Logarithhi of any one of 
the nine Digits is found with equal Difficulty for the 
firft. Therefore I Ihall give an Example in finding 
the Logarithm for the Number p. 

4. The Number 10, then, bemg already fuppofed 
at fuch a Diftance from Unity as contains 1 0000000 
equal Parts or RafiuncuUi which is the Logarithm 
or 10 ; therefore to find the Logarithm of -9, is to 
find bow many ^ or what Numier oi xhok RatiuncuU 
are contained between t and 9, which is done in the 
following Manner. 

5. Firft, make A=i, whofe Log. is 0.0000000 ; 
and B=:io, which Logarithm i. 0000000 ; as in the 
Table below. 

Secondly, between A and B, or i and 10, find a 
Mean Geometrical Proportional 0=3.16227775 the 
Logarithm whereof will be half the Logarithm of 10, 
l;/z. 5000000. 

Thirdly, becaufe C is much lefsthan 9, find an- 
other Mean Proportional D=:5.62 34 13 between B 
and C ; whofe Logarithm will be an yfritbmetical 
Mean between the Logarithms of B and C, vitt, 
0.7500000. 

Fourthly, becaufe D is ftill much lefs than 9, find 
another Geometrical Mean between B and D, viz. 
£=7.4989421,* whofe Logarithm is an Arithmetical 
Mean between thofe of B and D, viz, 0.87500000. 

Fifthly, fince E ii yet much Icfs than 9, find yet 
another Geometrical Mean F=8. 659643 2, between 
B and E ; the Logarithm of which will be an Arith- 
metical Mean 0.93750000, between thofe of B 
andE. 

6. And thus continue finding Mean Geometrical 
Proportionals between the Numbers next Greater and 
ne^ctLeJfer than 9, till you arrive to the Number 9 

itfelf. 



Chap.II. a Taile ^Logarithms* tt 

it/elf, which (hall be clear of all other Figures, but « 
Cyphers^ to the Number of Places propof^ ; which 
will happen at the twenty-fixth Trial, as you fee in 
the following Table. 

Proportionals. Logarithms. 



A i.ooooooo 

C 3.1622777 

B 10.0000000 

B 10.0000000 

D 5.6234132 

C 3.1622777 

B 10.0009000 

E 7.4989421 

D 5.6234132 

B 10.0000000 

F 8.659B432 

E 7.4989421 

B 10.0000000 

G 9-3057204 

F 8.6596432 

G 9.3057204 

H 8.9768713 

F 8.6596432 

G 9-305>204 

I 9.1398170 

H 8.9768713 

I 9.1398170 

K 9^Q579777 

H 8.9768713 

K 9^0S79y77 

L 90173333 

H 8.9768713 

L 90i73'333 

M 8.9970796 

H 8.9768713 



C 2 



o.oooooooo 
0.50000000 

I.OOOOOOOQ 

1. 00000000 
0.75000000 • 
0.5000000Q 

I.OOOOOOOQ 

0.87500000 
0.75000000 

I.OOOOOOOQ 

0.9375000Q 
0.8750000Q 

I.OOOOOOOQ 

0.96875000 
093750000 
0.968 7500Q 
0.953 1 250Q 
0.9375QOOQ 
0.96875000 
0.96093750 

0^953 '2500 
0.96093750 
0.95703125 
0.953^-500 
0.95703^25 
0.955078 u 
0.953 1 250Q 

0.95507812 
0.954IOI56 
0.95312500 



Ya Ofthi Method of making 

Proportionals. Logarithms. 



L 9017333? 

N 9,0072008 

M 8.9 9 707 96 

N 9^0072008 

O 9.0021388 

M 8.9970796 

G 9.0021388 

P 8.9996088 

M 8.9970796 

?) 9^^0021388 

Q^ 9.0008737 

P 8.9996088 

Q^ 9.0008737 

K 9.0002412 

P 8.9996088 

K 9.00024x2 

S 8.9999250 
p 8.9996088 



R 

T 

S 

T' 
V 

S 



9.0002412 
9.000083 1 
_8. 99992 50 
9.0000831 
9.0000041 
8.9999250 



y 9.0000041 

X 8.9999650 

S 8.9999250 

V 9.0000041 

V 8-.0000845 
X 8.9 999650 

V 9.0000041 
Z ' 8.9999943 

y 8.0000845 



0.95507812 

095458984 
0.95410156 

0.95458984 

0.95434570 
0.95410156 

0.95434570 
0.95422363 
0.954 1 01 56 

o 95434570 

0.95428467 

0^0542236^ 

0.9542 »4b 7 

0:95425415 
0.^5422363 

0.95425415 
0.95423889 
0.95422363 

0.95425415 
0.95424652 
0.95423889 
0.95424652 
0.95424271 
9.95423889 
0.95424271 
0.95424080 
0.95423889 

■ !■ ■ III! II ■!# 

0.95424271 
0.95424217 
0.95424080 
09542427! 
0.95424223 
0.95424217 



Chap.IL ^Ti?^/??/* Logarithms. 
Proportionals. Logarithms. 



»3 



V^ 9.0000041 

a 8.9999992 

Z 8.9^9994? 

V 9.0000041 

b 9.0000016 

a 8.9999992 

b 9.0000016 

C 9.0000004 

a 8.9999992 

c 9.0000004 

d 8.9999998 

a 8.9999992 

c 9.0000004 

c 9.0000000 

d 8.q99Q9Q8 



0.95424271^ 

0.95424247 

0.9^42422^ 

0.95424271 

0.95424259 

0.95424247 

0.95424259 

0.95424253 

0.95424247 

0.95424253 

0.95424250 

0.95424247 

0.95424253 
0.95424251 

0.95424250 



7. Thus, after a long and laborious Calculation, 
(confining of 26 Multiplications and Extradions of 
large Numbers) we at length have obtained the Lo- 
garithm of 9, viz. 0.95424251, which is cxacftly 
true to 7 Places or Figures : That is, if the Loga- 
ritbm^ or Diftance of 10 from Unity be fuppofed to 
confiff of 1 0000000 equal Parts, then the Diftance 
of 9 from Unity fliall confift of 9542425 of thofe c- 
qual Parts precifely % which therefore is the Loga- 
rithm of 9, becaufe it exprefles the Ratio of 9 to i . 
Jf the faid Logarithm, or Diftance of 10 from Unity 
were fuppofed to confift of looooo.ooooo.ooooo.- 
00000. 00000. oocoo. 00000. 00000 00000 .ooooo.- 
00000.000000 equal Parts ; then the Diftance of 9 
from i,wouldbe95424.25094.30324.87459 00558^- . 
06510. 23061.84002. 57728.38139.17296.59731^ 
of the fame equal Parts j and is its Logarithm to 62 
Places of Figures. But the Reader is not to imagine 
that this prodigious Logarithm was produced by the 
I fore- 



14 Of the Met hd of making 

foregoins operofe Method : No ; for tho* 'tis not 
impoffibie, *tis impraAicable by that Method -, but 
it is the Produd of modern Invention -, an Inftance 
of which will be given in due Place. 

8. Having thus found the Log. of 9=0.9542425 
if it be divided by 2, the Quotient or Half, 0.477 12 1 2 
will be the Logarithm of 3, by the Converfe of Ar- 
ticle 12. Chap. I. The Double of the Logarithm 
of 9 is the Logarithm of 8 1=1.9084850 by the fame 
Article. And thus by a continual Multiplication of 
this one Logarithm by 2, 3, 4, 5, (^c. you gain the 
Logarithms of all the Powers of Nine. 

9. 'Tis farther evident, that if the Ratio of 10 to 
I, 100 to 10, 1000 to- 100, 6?^. be fuppofed only 
Unity or I, as exprefled in the Table of Art. ij. 
Chap. I. the Logarithm of Nine, viz. 0.95424225 
will be jio other than a Decimal Fraftion, whofe De- 
nominator is the faid Ratio^ or Unity, with Cyphers 
annexed, thus tlUllll . 

^.8 *" 3=047712*2] g 






9=0.9542425 
10=1.0000000 

81=1.9084850 > 
100=2.0000000 

729=2.8627275 

1000=3.0000000^ 






j[-iP-««/. 10000000 
q=0.-Ali24ii 

J7"~"^* I 0000000 
10=1.0000000 

A — * •xooooooo 

100=2.0000000 

/^y— ^•looooooo 

^ 1 000=3. 0000000 

10. Hence 'tis obvious that the Logarithms are 
only the Decimal Parts of feveral Ratio^s^ i, 2, 3, 
&?f. of 10, 100, 1000, tfc. to Unity ; and that 
thofe Ratio's themfelves make the integral Part of 
the mixt , Decimal Logarithm, (as I may call it.) 
Thefe integral Parts of thofe mixt Decimal Loga- 
ritbmSy then are what we call the Indexes^ or rather 
Indices^ of thofe Logarithms, or fractional Parts \ 
wherefore, the Rations of lo, lOO, 1000, 1 0000, 
1 00000, &?r. to Unity ^ are the Indices of or belong-- 
ing to the Logarithms of all intermediate Numbers im- 
mediately 




Chap. II. tfTtf^Zf^ Logarithms. 15 

mediately above ihcfe Proportionals. So die Ratio of 
10 to I, viz. I, is the Index of the Logarithms of 
all Numbers between 10 and 100. And 2 (the 
Ratio of 1 00 to i) is the Index of all Lc^rithms of 
Numbers, from 100 to looo^ and (b on. 

11. From whence 'tis eafy to obferve, Thai the 
Index of any Logarithm contains a Number of Units 
lefs by one than is the Number of Figures in thaiNum* 
her which belongs to the f aid Logarithm. 

Therefore f g^. 

12. Now as this Index is thus called, from its/x- 
Heating or fhewing how many Places of Figures 
there are in the Number of the Logarithm ; fo, oa 
the contrary, the Number of the Logarithm, as 
plainly ihews what the Index of the Logarithm muft 
be. And therefore, fince in the Tables ofLogarithms, 
all the Numbers of thofe Logarithms are appofitely 
placed in proper Columns by them, *tis entirely need- 
Icfs to print the Indices^ as being well enough known 
by the Numbers ; and thus they are (for this Rca- 
fon) omitted in feveral of the faid Tables ; and con* 
fequently in thofe we have here, info concife a Fotm^ 
now firft of all made publick. 

13. What has been faid hitherto, relates altoge- 
ther to the Logarithms of Whole Numbers ; but the 
fame DoStrine is equally applicable both in Theory and 

PraSice^ to fraaionai Numbers \ for in EfFeft, the 
Properties of Decimal and Jin;>ole Numbers are the 
feme. Thus the Series of decuple Proportionals ^ i, 
10, 100, 1000, 1 0000, 6fr. may be continued as 
well belowVviitj^ as above it -, for the following Rank 
of Numbers, viz. rdurs, ttoE^j irs* /ny> i» i^t 100, 
I GOO, 10000, i^c. which may be thus exprefs'd, 
0,0001, 0,001, 0,01, 0,1, I, 16, 100, 1000, 

lOOOO, 



1 6 Of the Method ofmakwg 

loooo, lie. are all in a Geometrical Decuple Ratio 
tr Preportion 5 and confequently equally diftant from 
Unity on each Side, and io their Logarithms eqiii- 
differcnt, and the fame. The Ratio's then, in the 
Scale above Unity, may be called Pofitive Ratio^s^ 
and thofe below Unity, Negative Rations i and thus 
the Index of a Logarithm of a ^i&^/^ Number^ or /«- 
ieger^ fcall ht fcfiti'oe^ but the Index of the Loga- 
rithm of FraSian ftiall be negative^ 

14. Andiince Vulgar FraStions^ Duodecimal^ Sex- 
cgefimaU &c. Fraftions, are all reducible to Decimals ; 
*tis evident that all Numerical Arithmetic whatfoever 
is fnbjeft to, and manageable by this Art of Logarithm 
mical Arithmetic. The only Difficulty being in ad- 
apting or readily finding the proper Index to the 
Logarithm of a Decimal Number. For, fince the 
Index in the Logarithm of Integers^ fhews only the 
Number of Places or Figures m the faid Integers, *tis 
plain thofe Indices decreafe^ as the Places of Figures 
in the Integer decreafe, and intirely vanifli when 
thofe Places of Figures become Unity or one, that 
Js, the Index is then o ; and confequently canndt 
ferve for Decimal Numbers. 

15. Therefore fome new kind of Indices or Cha- 
^ raSlerifiics muft be invented, which ihall be proper 

only to Decimal Numbers^ as the other are to Inte* 
gers ; and fuch as (hall as readily difcover the Num^ 
ber of Cyphers to be prefixed to thcjignificant Figures 
m the Decimal^ as the other determine the Number 
of Figures in the Integer^ are in the fame degree ufe- 
ful for Decimals^ as they for Integers. For as to the 
figi>ificant Figures of the Decimal^ the Logarithm it 
felfdiCcovtrs them ; all therefore that is farther- ncceA 
iary, is to procure fuch Indices as fhall at all times 
denote how far ^ or how many Places froei Unity ^ the 
firfl Jignificant Figure of the Decimal muft ftand ; or, 
which amounts to the fame, how many Cyphers mut 
be prefixed to compleat the true Value of the Decimal. 

16, 



Chap.lL a TaiU '^Lo GAR ttuMil t^ 



c^y 500=4.6^^66936 
Integers^ 4750=3.6766935 

t 4';S=2.6y66g^6 
47,5= 1.6766936 
4,75= 0.676693^ 

yb475:=z.i. 6766936 

y0047$=:.7. 6766936 

^0004.7 5=2.6.6766936 

,00004.75=. $.6766936 



It 



& 



16. Tht Indices {qx 
this JPulrpofe am there- 
fore judg'd beft, Wi&iVi? 
i^/j8f^ fubfiraSted from 
9, ^/i// i&aw a Re 
mainder expreffing the 
Number of Cyphers to 
be prefix" d'^ thus the 
Index o( a Decimal 
who(e firft Figure (to 

the Left,)isfignificant, i 

ttiuft be 9, becaufe 9 — 9=0, tha^ is nd Cypher is ta 
be prefixed* ^ And thus, a Decimal thaj; has i^ 1, 3^ 
4, &ff. Cyphers^ muft have the Indices^ accordingly 
8, 7j 6j 5, fcf^. becaufe 9 — 8=1, 9 — 7=2*9 — 6=3, 
9 — 5=4, (^c. denoting the Cyphers to be prefixed to 
the Decimal Numbers. This is tftridcnt by the Ex- 
amples in the Table above ^ but it is proper to dott 
thefe new Indices on each Side^ to diftinguifh 'em 
from thofe which belong to the Logarithms of//r- 
tegers. 

17. From all that has been faid hitherto, *tis evi- 
dent, that while th^Jignificant Figures in af^ Number 
ivhatever^ Integral or Decimal, are the fame, the La-' 
garithm of thofe Figures will be the fame alfo ; the 
Variation occaiion'd by the differing Nature of the 
Number, being only in the "Indices^ as denoted ia 
the faid TaBle. But fomething farther will be faid 
of the Nature znA Variety ofhdices^ when we com^ 
tojireat of the PraSiical Rules oi A^diti/>n and Sub^ 
firaSion of Logarithms ; I fliall now proceed to the 
next Chapter, wherein the ^erious iVatun^ or Doc- 
trine of Logarithms will be farther illuftrated, and 
rendered more obvious to the Scnfcs by Geometrical 
Schemes and Demofffirations. 



D 



CHAP. 



i8 Vfle Nature and Properties 



CHAP. III. 

iTbe DoSirine of the Nature and Properties ofLo^ 
garitbms farther explained and illujirated^ bf 
means of tbe Log ARiTUMic CvRvt. 

i.T N explaining the Nature of Logarithms from 
X the Logarithmic Curve^ *twill be expedient to 
Ttprtknt Proportional ^antities by Letters or Species ; 
fis being fuited to a more univerfal Theory than Ntm- 
bers^ and better applicable to Right lAnes^ by which 
alone both Numoers and their Logarithms are repre- 
fcnted in the Geometrical Method now before us. 

2. In a Series of ProportipnalSj then, increafing 
from Unity, let the firft be ==tf ; then will the Se- 
ries be I, ^% a* J a\ a\ tf% &c. where /tis evident 
the Indices or Exponents of the feveral Powers of tf, 
are a Series of Numbers in Arithmetical Progreffion^ 
each, whereof fliews the Place or Diftanee of its TVnw 
from Unity i thus the Term a^ is (hewn by its Index 
4 to be in the fourth Place from Unity ; and a^ is in 
the Jifth Place ; or a^ and tf' are four ^nAfive times 
snore diftant from Unity than the jf^^2Vr»» a ; which 
is her^ the common Ratio alfo. 

3. From hence you obferve, that the Exponents 
of the Powers of the Terms in the Series i, a*, a\ 
a\a\ a*^a\ a^j &c. are the Logarithms of thofe 
Terms rcfpedively. Let the Exponent be =^ ;. then 

c the Terms, i, «, aa^ aaa^ aaaa^ aaaaa^ aaaaaa^ Sec. 

I theExpon. o, ^, 2^, 3^^ 4^?, 5^, 6e^ &c. 

Therefore 

c As aaxaaa=a* 5 aya'=a^ ; and aamacf^zra'^^'a^ 

|Sq 2^4-3^=5^; H-5^=:6^ ;.and 2e+3ie=zgie. 

and fo every where, agreeable to the Nature of 2>. 

garithms 



Chap.III. g/LoGAniiTHMS explained. 19 

garitbms beibre defcrib'd, in the two precedingChapv 
ters. 

4. If between i and a there be put one mean Pro- 
portional^ viz. ^?j its Index ot Exponent muftbe 
wrote i ; thus ^a:=:a^ \ for o, i, i, have an Ariib^ 
metical Ratio ; and fo the Terms i, ^, a^tat Geo* 
metrical Proportionals. Thus» a mfm Proportional 
between a and tf^, is ai^assa^i=o^. Al(b if you 
conceive two mean Proportionals between i and a^ 

they fhall be a^^ a^ ; for i : a^ :iinia\ ando, i, $9 
I, have an equal Difference. 

5. Moreover the feme Series of Geometrical Pro* 
portionals may be continued both ways, and be made 
decreafing as well as increajing ; that is, it may as 
well defcend below Unify to the left Hand^ as afcend 
above it to the Right. Thus the Terms ^t, ^i, ^t, 
^iy 49 I9 ^9 ^% 41% tf^ ii% &c. areall in theiame 
Geometrical Progrejfion. And fince the Diftance of 
a* from Unity towards the right Hand is pofitive^nnd 

denoted by the pojltive Exponent -Ht ; fo n^ being 
equally diftant on the contrary Side^ or helot» Unity^ 
and this Diftance being negative^ therefcH^ the Ex- 
ponent thereof may be thus negatively expreis*d, ' 

a^^ z=a:^ I, and fo d^ is the fame as ^~^ ; thus the 

Series tf~^ jJ~^ iT^, a~^^ a~\ U «» 4% a\ 
a\ ^S &c. is the fame as the laft foregoing. 

6. Therefore, in thoie 5ieries, if a repreient any 
Right Line (fee Fig. i.) aaj or the Square thereof, 
is not to be taken as a C^antity of two Dimefj/lons^ or 
Surface^ viz. a Geometrical Square ; but only as a 
Line that is a third Proportional to fome Line taken 
as Unity ^ and the Line a. So likewife aaw^ or aaa^ 
is not a ^antity of /i&r^^ Dimenjions^ or a Geometric 
cal Cube 5 but a Line that the fourth ^erm or Pro* 
portional in a Geometrical Series^ whoie j&;t ?Wtii is i * 

D 2 and 



5.0 The Nature and Properties 

and fecond a. And thus you are to conceive of al! 
the reft; viz. limply as Proportionals of Length 
pnly. 

7* Thi^ bein^ premifed t|ien, if on any Line, as 
AN (Fig. 2.) both ways indefinitely extended from A, 
be taken the equal Diftances Ae=BCE=EG3=GI=3 
ILsLN, to the right Hand \ and Acs=ce=eg, ^c^ 
on the left ; and if on the Points g, c, c, A, C, E^ 
G, I, L, N, be ereifted to the Right Line gN, the 
Perpendiculars gh» ^, cd, AB, CD, EF, GH, IK^ 
LM, NO, which I^t be vtx a continual G^ciw^/r/V^/ 
Proportion to each other, and reprefent Numbers^ 
whereof AB=i, or Unity. The Tops of thefe 
Lines being duely join*d, from what is call'd the 
Logaritbptetic Curve^ viz. hBHO 5 by which we 
are farther to explain and illuftrate the Nature and 
Properties oi Logarithms. 

8. But a more com pleat Idea of the Logaritbmical 
Curvey may be conceiv'd by a twofold Mption of the 
Line AB ; the one equable^ the other accelerated or 
retarded in any given Geometrical Ratio : For Ex- 
ample, if the Right Line AB moves uniformly along 
the AN, fo that the End A defcribes'^^w/j/ Spaces 
AC=:CE=?:EG, Gf r. in equal Times ; while in the 

• fame Time the faid Line AB fo increajes^ that the 
Increments thereof generated in equal 'Times ^ be fro^ 
pcrtional to the whole increafing Line ; that is, if AB 
in moving forward to <z^, be increafed by the Incre^ 
ynent ob^ and in an equal Time^ when it comes to CD^ 
the Increment thereof is jyp ; and Dp : ^^ :: ^^ : AB ; 
then the End B of the faid Line AB thus continually^ 
increajing or decreajing in xhtfame Ratio^ will defcribe 
the Logarithmic Curve. For fince AB : ko :: ab : Dp 
:: DCidq^ &c. it Ihall-be, by Qompofition of Ratio, 
a3 AB lab y.abi DC :: DC : dc ; and fo on. 

9. Since (from wjiat has been faid) the Line 
AB=i , and the other Lines CD, EF, GH, iSc. pro^* 
cced ftom theiKC to increase in a Geometrical Ratio^ 

and 



Chap. III. ofh OGARiTHMS explained. %i 

and their Diftances AC, CE, EG, Csfr. are all efual 
to each other (by Art. 7.) Therefore it follows that 
the Diftances or Lines AC, AE, AG, ^c. are the 
Logarithms of thofe Numbers reprefented by the 
Right Lines CD, EF, GH, 6?r. according to tho 
Definition of Logarithms ^ Chap. I. Art. i, and 13. 
For if AC=i, then AE=2, 'AG=3, (<fc. and fo in 
the Serits of proportionate Numbers AB, CD, EF, 
GHj IK, LM, NO, l^c, we have the Logarithms 
o, AC, AE, AG, AI, AL, AN,j., 
o,iAC, aAC,3AC,4AC, 5AC, 6 AC, J^^* 

10. As Logarithms^ then, are the Exponents of the 
Rafters of proportional Numbers to Unity in any Se- 
ries 5 or ihew the Place^ Power ^ or Order of the. 
Numbers with refpeft thereto; 'tis plain that the 
Logarithm of AB or Unity muft. be =^, becaufe U- 
nity is not diftant from it felf. Alfo if the Ratio of 
CD to AB be z=:x^ then fhall the Ratio of the Num- 
ber EF to AB=x X K=xx ; or Duplicate of the former 5 
and the Ratio of GH to AB=xxxx;if=;f', or Tripli- 
cate of the firft Ratio x. Thus the Ratio of NO to 
AB=i;f^/ And confequently the Numbers^ their Ra^ 
tio^s^ and Logarithms^ will ftand in the following 
Order, viz. 

Prop. N°. AB, CD, EF, GH, IK, LM, NO, 6fr. 
Ratio's ... I, AT, ;v*, ^% x% ;?% x^^ tfr. 
logarithms o, AC, 2 AC, 3 AC,4 AC,5 AC, 6AC,&?f . 

11. If four Numbers be fuch, that the Diftance 
between t\it firft and fecondy be equal to the Diftance 
between the third and fourth, (be the Diftance bt- 
tyvtcn the fecond and third what it will) then thefc 
Numbers will be proportional ; for let the four 
Numbers be AB, CD, LM, and NO ; then becaufe 
AC=LN, it will be AB : Dp :: LM : OT, by Art. 8. 
Therefore (by Compofition of Ratio) AB : DC :: LM 
: NO. And thus of any other four Numbers in the 
Geometric Serie^^. The Converfi alfo of this Propo- 
iition i? a3 inanifeftly true. 

12. 



22 ^['be Nature and Properties 

12, Since ABrai, and its Logarithm =o ; and 
fince Unity is to the Multiplier as the Multiplicand is 
to the Predulf, in every MultipJication ; therefore to 
every Addition of Logarithms j there correfponds a 
Multiplication of their Numbers. Thus, 

Number, CDxEF=:GH ; and EFxIK=NO, 
Logarithms,AC+AE=AG5andAE+AI=AN. 
And in Divifion^ fince Unity is to thtDivifor as the 
Siuotient to the Dividend % then (by -/fr/. ii,) for 
every SubfiraRion or Difference of Logarithms^ there 
correfponds zDivifion of their Numbers, Thus, 

Numbers, GH-r-EF=CD; and NO-^EF=IK. 
Logarithms, A(ir-AE=AC ; fo AN— AE=AI. 

agreeably to what was fhewn in Chap. I. Art. lo. 

i^. Again, fince Unity y any affum^d Number ^ the 
Square thereof, the C«^^, the Biquadrate^ &c. are all 
continual Proportionals^ their Diftances from each 
other will be ^jf«^/ to one another ; and are therefore 
as the Numbers AB, CD, EF, GH, 6f(r. in the Se- 
r;es. Confequently if the Diftance or Logarithm of 
the Number CD be multiplied by 2, 3, 4, fcfr. there 
will anfwer an Involution of the faid Number to the 
Square^ Cube^ Biquadrate^ &c. Power. 

Thus for the Square 
Numb. CDxCD=:EF ; and EFxEF=IK. 
Log. ACx2=AE5fo AEx2=Al. 

For the Cube 
Numb. CDxCDxCD=:GH -, and EFxEFxEF=Nb. 
Log. ACc 3 = AG ; and fo AE x 3 =AN. 
For the Biquadrate 
Numb. CDxCDxCDxCDC=EFxEF)=IK. 
Log. ACx4=: ( AEx2=:) AI. 

Which is the lame as was obferved in Chap. I. 
Art. 12. 

14. If the equal Diftances AC, CE, EG, GI, IL, 
be bifcficd, and in the Points of Bifedion /ar, r, e^g^i^ - 

there 



Chap.in. of Log Animus explained, aj 

tiiere be erefted the Perpendiculars ai^ cd^ //, gb^ ik i 
thefe, by means of the Curve, will be all Proparti^^ 
nals ; and the Number LM will be in the tenib 
Place from Unity or AB : if then we put LM=io» 
and fuppofe its Ratia to Uniiy be looooooo, fudi 
RatiuncuUf as that of n ^ to AB, or Unity b i. 
iThen will the Numbers and their Logariibms be «• 
exprefi'd in the following Table. 





Numbers. 
AB=i 


Logarithms. 
0. 




ab =1.259, t^c. 


Aa = loooooo 




€0 = 1.585, £s?r. 


AC= 2000000 


' 


cd = 1.996, fcfr. 
EF =2.512, 6f^. 

ef :=: 3.163, (^c. 
GH= 3.982, £sfr. 


Ac = ^000000 
AE= 4000000 
Ae = 5000000 
AG= 6000000 




|i& =5.072, fcfr. 
IK =6.310, &ff. 


A^ = 7000000 
AI = 8000000 




i* = 7.944, 6fr. 
LM=io. 


At = 9009906 
AL =10000000. 


i 










fJumiiers. 


Logarithms. 






AB = I 









CD = 2 

EF =3 


AC= 3010300 
,AE= 4771212 






GH = 4 


AG= 6020600 








IK =5 
LM= 6 
NO= 7 
PO = 8 
RS =9 
TV =10 


AI = 6989700 

AL= 778151^ 

.AN= 8450980 

AP = 9030900 

AR= 9542425 
AT =ioooooQO 





15. If BKV (Fig. III.) be the Logaritbmetie 
Curve, and AB=i, TV=io, or 10A6; and all 
Ae intermediate Lines CD, EF, GH, fs?r. be as 
Digits 2, 3, 4, fcf^, to IP i thefe Lines will befitu- 

ated 



2 4 iie Nature of LoGAHtTUMs. 

ated at unequal Diftances from each other ; and fo 
€titit Logarithms ACy AE, AG, fe?r. not proceeding 
to increafe by equal Differences, (hew the Numbers 

AB, CD, EF, 6f r. are not ordinately (or all of them) 
io a Geometrical Ratio or Proportion. If the Ratio 
of I o to I, or TV to AB be fuppofed to confift of 
looooooo RatiuncuUj and confequently the whole 
Logarithm AT of iooqoooo equal Parts ^ then the 
Diftances of each of the nine Digits from Unity, viz^ 

AC, AE, AG, £sfr. fhall confift of fuch Numbers 
of thofe fmall equal PartSy as are expreffed in the 
Table, oppofite to the faid Digits. 

-1 6. Yet fince fome of the Digits are in a Gebme^ 
trie Ratioy as 1,2, 4, 8, their Logarithms will be 
equidifferent ; fo AC==CG=GP ; and becaufe 1:3:: 
3 : 9, there is AEssER ; and thus it appears that 
tho' the nine Digits are not all in continual G^(?iw^/r/- 
cal Proportion^ yet fome of them are fo ; and the 
reft are fome of thofe Proportionals^ of which there 
be looooooo between AB=i, and TV.=io. If 
the frfi Term from Unity be called Xy the fecond 
will be x*y the third ^% &c. and fince. the Number 
10 is the 1 0000000th Term of the Series, it will be 
TV=^'°°^°°°^=io, Alfo CD=x^°'"'"°=2; and 
EF=^"''"*"=3 ; and fo on. Whence every Digit 
is fome Power of that Number which is th& firft Pro- 
portional from Unity : The Exponents or Indices of 
the Powers being the Logarithms of the Numbers^ 
agreeable to Art.'io. of this. 



^ 



CHAP. 



C H A P. IV. 

Tie Nature of Logarithms and their Indiceil 
ivben the Numbers are FraStionSy farther ex^ 
plairid by the Logarithmic Curve. 

i.^TTE have hitherto principally confidci*cd thft 
VV Nature and Properties of Logarithms of 
vohole Numbers or Integers^ and have obierv^d that 
in the Decuple Series t, io, loo, iooo,,€^f. the 
Terms have their Rations to Unity iMrmative or pth 
fitivCy viz. I, 2, 2, 6ff. orthu$, \i^ 4-a, H-j^ 
&r. the contrary of which happens when tne Num- 
ber of the Logarithm is not integraU but fraHional^ 
or cxprefles only fome/ri?^i^»tf/^ Part of Unity. FoJf 
there the Series beiilg continued on the other Side^ or 
below Unity ^ hzxh tht Indices of the Powers of the 
Terms of a polity direftly oppqfite to the former | 
. And therefore as tbofe Indices were pcfittve^ to tbefe 
will be of a negative Nature^ and import the Terms 
to be below the State of Unity ^ or rather of integrity i 
and willbeafieded with the' Sim --»» as>«»i» — ^2^ 
— 3, 6?r. as is evident Chap. IIL Art. 5. 

2.. Wherefore fince AB reprefents Unity ^ (Fig. 11.) 
all the Numbers in the Series towards the right Hand^ 
or above Unity ^ CD, EF, GH, fcfr. being integral^ 
and having the Ratio greater than Unity ^ will have 
the Logarithms thereof pqfitlve^ viz. -f-AC^ -l-AEt 
+AGp &c. But thofc Numbers or Terms on the 
Lefi^ or bck)W Unity, cd, ef, gh, 6?f. being /w^- 
tionate^ and having the Ratio lefs than Unity or if- 
creafingy will have the Logarithms thereof Negative, 
V/z. —Ac, -—Ac, -— Ag, €g?r. Andfothei;»;i;V^j 

E qI 



26 Tie Natuf^ebf Log Ai^iTfiMS^ an4 

ofthofe Logarithms will be refpeAively affirmative 
or negative. 

3. Since then (as hath been all along (hewn) the In^ 
dices of the Logarithms of Integers^ as bemg affirma^ 
tive^ muft be addei^ that fo their ^um may indicate 
the PrcduEt of the Numbers multiplied ; as AC+AE 
s=AG, which fticws that its iV«iwii?rGHfi=sCDxEP, 
the Numbers multiplied ; fo if the Index of one Lo- 
garithm be negative^ and tht Index of another htaf- 
firmative^ the Difference of thofe two Indices muft 
be taken for th& Product of the Numbers. Thus to. 
multiply the Integer GH by the Decimal cd, their 7»- 
^/V^j being H-AG> and — Ac, their Difference 
AG — AssAE, and fliew the Number EF is the 
ProdflEt of the other two. And here, becaufe the 
greater Index GH is affirmative^ the Difference alfo 
AE is affirmative^ and the Product EF to be an In- 
teger or on the right Hand of Unity. 

4. But if the Decimal gh be to be multipliedhy the 
Integer CD, whofe affirmative Logarithm or Index 
AC is the negative Index of the Decimal^ viz. 
Ag» their Difference Ag — AC = Ae, is negative 
alfo, and fo ihews the ProduSltf will be a Decimal^ 
or ^e/^w Unity. Again, if both the Indices of Loga- 
rithms^ whofe Numbers are to be multiplied^ are ne- 
gativej their 5tfw ftiall be a negative Index whofe Lo- 
garithm points out the ProduSi (in this Cafe) always a 
Decimal^ or in the Series Wewc; Unity AB, For Ex- 
ample, to multiply the two Decimals cd, ef, the Sum 
of their negative Indices Ac+Ae=Ag is negative^ 
and fhews the Decimal Pr^dullgh^ ever below Unity. 

5. The Reafon of all which is very plain; for 
fince Unity is to the Multiplier as the Multiplicand to 
the ProduB^ and the Logarithm of Unity is =0 ; 
Unity ^ any /te;^ Numbers and their ProduSt^ together 
with their Indices^ will ftand as follows : 



Chap.IV. their Indices^ farther txplami, 27 

, cind. 0, AC; AE, AG, 
'• iNum.AB:CD :: EF : GH, 



'1 



M 



iCD X £F=GHx i=GH. 

o, —Ac, +AG, -l-AE, 
. AB : cd :: GH ; EF. 

o, '+-AC, — ^Ag, — Ae, 
AB : CD :: gh : ef» 
♦i«H^fo«. C AC— Ag = Q — Ac = — Ae, 
thcrcfort JcD x gh = i x cf = cf. 

CO, —Ac, —Ac, — Ag, 
♦•t AB : cd :: cf : gh, 

therefore }^?^^^ = ^~\« = -A& 
C ef X cd =s I X gh = gh. 

6. FrQm whence 'tis evident, if the Numbers are 
hotb Integers^ the Produ£I will fall in the Scale above 
for be greater than) eithpr ; if they be of different 
Sorts^ the Produff will fall between them, above 
Unity if tht greater Number be an Integer^ orbehw. 
Unity ^ if it be a Decimal. Laftly* if both the Num* 
bers be Decimals^ the Product will fall below Unitj 
and them botb^ or will be lefs than either of them. 
Thefe things well obfitrv^d and underftood^ make all 
the Myftery of the 4ritbmetic of Fraai(ms by I^^- 
r;/ibij vaniih, where it is taught with the Ufe ofne^ 
gative Indices to Logarithms oifraStional Numbers^ 
as it is to be found in fome Books. Note^ \ have all 
along in this Chapter, faid affirmative or negative In* 
dicesy and riot affirmative and negative L^aritbms i 
becaufe the Indices only (hew the ^ality of the 
Numbers^ viz. whether above or belowUnity^ greater 
or leffer than it, i. e. whether they be Integers^ or 
wholly Fra£f ions i and therefore theCe Indices iqvA be 
carefully added or fybjiraj^edy a^ o^cafion requires. 

E 2 and 



90 ^ ffature ff Lo g Ami t h m8, and 

and as before direAed ; but the Logarithms fhewing . 
th^ Difiances o( Numhrs from Unity indifferently^ 
or without rcfped to thtOrder oi akov^ or helow^ are 
always to be. added. 

7. But fince this Method of negative or different 
Indices to Logarithms is attended with Mditiphj Sub^ ^ 
JiraStion^ and other intricate CautionJi peculiar to 
themfclves^ it can't be recommended Co much asan^ 
other Method more approv*d^ and therefore more ge^ 
nerally ufed\ whefein th^ Indices of all I^ogfritbms^ 
(both offraSional ^ well as inSegral Numbers) un- 
dergo the f^meco^xnon ^oMagement with their X0- 
garitbms,\ and fo can*i he fo difficult to Learners as 
the £?/i&^r ; tho' it is alfo attended with particular 
Rules ^ as you'll iiqd farther on, and i$ as foUows : 





Nimibers. 


Indices. 


Lor- 1 


ut =,0000000001 


— 10 90.0000000 




rs = >ooQOOOooi 


tr = — 9 I. 91.0000000 


'' 


pq=: ,00000001 


tp = — 8. 2. 92.0000000 


1 


no =2 ,0000001 


tn = — 7. 3. 93,0000000 


lm=;;: ,000001 
ife =; ,00001 


tl=: — 6. 4. 94.0000000 
ti=; — 5. 5. 95.0000000 


^ 


gh:;;: . ,OOOl 


tg=: — 4. 6. 96,0000000 




efss ,901 


\:e=-^ 3, 7. 97.0000000 




cd=;, ,01 


tc== — 2. 8. 98.0000000 




abs 9I 


ta=::-T- I. 9. 99.QOOOOOO 


1 


ABs? I 


t A 5= 1 0. 1 00.0000000 


I 


CD:;; 10 


tC = + 1 . 1 1. lOI.OOOOOOO 


EF= 100 


tD=;4- 2.1 2. 102.0000000 

m 



8. Let AB be ==?, or Unity (F%. IV,) and let 
tu be ifraSlionai Number ip times as far below U- 
fiity A3B^ as CD is ab&ve it ; then will AtzziqAC, 
ima fiippofing the Number CDs=.io, ^is evident 
te;5iooor6QQQ0Q> ^ f0999OpOO0i J if then, inftea<l 



Chap, i V. their It^ctSy farther explained. 39 

of the Logarithms beginning at Unity AB (as in 
the former Cafe) ^i^e make 'cm beg^ from tu= 
j^coooooQOOi, then the /»</^x of L^arithm of the 
praaim rs^ will be i 5 of /y, 2 1 of i»» 3 ^ and ib 
of Unity AB> 10 ; and iinceCDsio, £F=xoo ; the 
Indices of their Logarithms will be tC=5ii, and 
tD=i2, ^c. as is evident in the foregoing Table ; 
wh^you ke tht fraaicnal and integral Numkers 
with Indices oi three feveral Kinds fitted to the Lo-» 
garithms of the faid Numbers. 
• 9. That every thing might be made plain, I have 
made the Index of the Lc^arkhm cS Unity ^ zsOt in 
thtfirfi Rank of iHdiees ; and according (as we luive 
iben' above) the Indices of all the Logarithms of 
Numbers ab$ve AB will h^pqfitive^ as -hi, -Hi, (^c. 
and of thofe below it negative^ as — i, —2, —3; 
6f r. as in the Table. But if (as in the fecond Rank 
of Indices) the Index of the L-ogarithm of Unity AB 
be made =10 ; then (ball the Logarithms begin (as 
before fuppofed) from the loth Place ielaw it, viz. 
from the Number ut=,ooooooqoo I, sind then all 
the Indices of all t\x^ /rational Numbers between 
f bcoooooooo Of 0,0000000001 and I, will be ^r- 
mative OTpJitivCy and not negative as before ; as is 
fcen in the Table. So that by this Method we have 
got pofitive Indices for the Logarithms of Decimal 
Numbers as well as for Integral Numbers ; and fa 
both ca{)able of the fame Management, which could 
not be with negative Indices^ as before fhewn. 

10. Alia in the third Rank (which is^ here gtveti, 
not for Variety only, but hecaufe 'tis oftentimes of 
Ufe) the Index of the Logarithm of Unity AB is made 
=100 ; and confequently that thefe Logarithms be- 
gin from the FraStion which is the 100th Proportion 
nal below Unity, in the Ratio of r to 10, or -ti to u 
Whence the Indices of Logarithms of all FraSiions 
between o, oooBoooooooooooooooo 0000000000- 

9OOOOQOOQOQOQQOOQQOQ OOOOOQOOOOOOOOOOOOOO 

OOOOOOOQOO 



30 The Nature ofLo GAR i t h m s, anJ 

oooooooooooooooooooooooooooooi and i, will 
of courfe hcpqfitive ; and fo in thisClafs, the Indices 
oiLogaritbms of all decimal Numbers (greater by far 
than^, or evenCuriqfityj candefire) will bepofitive 
alfi), as well as thofe of the Logarithms of integral 
Numiirs ; and therefore in (hort, we have obtained 
fqfitive Indices for the Logarithms of all Numbers in 
general^ and fo the Trouble of Addition and Subjirac^ 
Hon di different Farts of Logarithms at the fame 
time, is avoided, as was propofed. 

11. But (as I have before obferved, Chap. 11. 
Art. 1 6.) the Indices of the Logarithms oi Decimals 
ihould be diftingui(hed from the Indices of Loga- 
rithms of Integers in Operations ; and this I think is 
beft done as there dire&ed, viz. by fixing a Dott on 
each Side the faid Indices. And thus manifeftly ap* 
pears the Reafon of all that is delivered in Chap. 11. 
from Art. 13. to the End. 

12. Now in multiplying^ dividing^ &c. of Frmc- 
tionsy the FaSors may be either both Decimal ; or one 
DecimaU and the other an Integer \ if both the Fac^ 
tors zrt Decimal^ the ProduSl will bt Decimal^ in Mul- 
tiplication. If the FaSors be of different Sorts y and 
the Ratio of Unity to the Decimal be either lefs^ 
equal /^, or greater thap the Ratio of Unity to the /»- 
teger ; then ihall the Produdb be greater^ equal to^ or 
i^s than Unity \ i. e. it will be Integer^ Unity, or De^ 
cifhaly in multiplying ; all which will be evident by 
the following Examples^ ferving as fo many Rules 
for the right ordering of Indices m Operations. 

13. TAB : cd :: ab ; ef ") «^- T cd x ab=i x ef=;:«f. 
\ tA. .tc. .ta. ,te. > % v.tc.-f-.ta.=tA-+-te. 
^'i 10. .8. .9. .7. J-S j.8.-K9,=io.'+-.7. 

CThat is . , . . . 8,.H-.9. — io.==:.7.==: 

Index of ef. 



Chaf). IV. their Indices^ farther explained. 3 1 

ab :: EF : CD! ^ f abxEFsi xCDsCD. 
tE. tCA § ^ .ta.-HE=tA.H-tC. 
12. ilJ-S L.9.+i2.=io.-t-ii. 

nfcqucndy 9.+12. — 10=1 i.ae 

Index of CD. 




rAB:ab::CD:AB1u: fabxCD=ixi=i 
3 tA. .ta. tC. tA. S g ^{ .ta,-HC=tA-f-tA. 
3-^ JO. .9. II. 10. J -S 1^.9.-4-1 i=icH-io. 
^Therefore »•..<♦..* .9.+11. — losssios^ 

Index of I. 




Index of ab. 



14. From thefc Obrefvations Well cOhfider*d, •tis 
cafy to apprehend the Truth of what is delivered in 
Art. 1 2 . above. And Unce AB, in thefc Examples of 
the ProduRSy hath for the Index of it8 Logarithrti lO^ 
*tis equally obvious what the Indices of tht Logarithms 
of the Products would be, wert the (aid /xr^^;^ of the 
Logarithm of AB made 100. In iHtixsprefent Cafe 
were tAsssio, if we rejcft 10, the Indices of the 
Logarithms of integral ProduSls will be the fame as 
if the Logarithms began at AB or Unity 5 as is plain 
in the 2d and 3d Examples \ and accordingly if 
tA=ioo, we rejeft 106, and the Cafe is the fame. 

15. But fince 'tis moft convenient to have the In- 
dices of all Logarithms of integral Numbers to begin 
from Unity in the fimple Order, o, I, 2, 3, (^c. as 
if the Logarithms did really begin from thence; fo 
*tis but rgedting 10 from the /aid Indices In their 
prefent State, dnd what we defire is obtained. 

Thus 



32 TbeNaiun df Log AJBitTUMs^and 



ft. 

C* 1 2. 



•J* ^ .9. — lO ar ,7, 

Thus the forc*3 2. .9. -h 12. — • 10 «= 1 1. 
going Cafes 1 3. .9. -f- 1 1. — 10 = 10. 
(^4. ii*-f-. .8, — 10 =zr .9. 
by rejedHng 10 from r.8. -+- .9* — o = .17. .7. 
the /»ifV^i of thej.g. •+- 2 — '0= ix. i* 
Logarithms of/»-i .9. + i •^- o =5 10. o. 
tegers^ becomes (, i 4- 8 — o :^ ,9. .9. 

Then from the Sums •17, 11. 10. .9. again rqefting 
10 (where 'ti^ found) we have the Remainders .7. i. 
6. .9. the true Indices of the Logarithms of the Pr^-* 
^ir^jy tsrequiredi 

16. If either PaSloTy or the ProduSl of them^ 
exceeds the Limit of ,o'oo>oooooo » <^r is lefs than 
^ooocoooooi, we (hall find it moft convenient to ufe 
the Indices of thofe Logarithms of which the Indeoi 
of the Logarithm oi Unity or AB is =-100, viz* 
thofe -Indices in the 3d Clafs in Table of Art. 8. here- 
of. And if, in all the foregoing Cafes, inftead of r^- 
jeUing 10, we now rejeft 100, we (hall have the /«- 
dices of the Logarithms of the Produds the fame 
kind as before. 



Thus^ 



byrejcfting 100, 
becomes 



.98. + *99. — 100 = .97. 

.99. 4- 102 IQO = lOI. 

'99' + ^^^ — ^00 =5 lod 
loi.-f- .98. — 100= .^g. 

.98. -f* ,99. ' — o =.197. 
.99. + 2 — o = lOI. 
•99* + I — o = loOi 
loi. 4* •98. — o = 199. 



•9^ 
I. 

G. 

*99- 



Thus here you fee the EiFe<3: the fame as abov^t 
Art. 15. if .97. .99. be deduced from 100, as .7. 
.9. were from 10, the two middle Indices being the 
fame in both Cafes ^ 

?7- 



chap. IV. their IndkeSy farther explained. 33 

J 7. In Bivifion^ the Divifor is to Unity as the D/W- 
d^nd IS to the ^otient ; arid fo the Diftance between 
Unity and the Divifor is equal to that: between the 
Dividend and patient. If then the Fradion ef be 
divided by ab, becaufe aA is :sec, therefore cd is the 
^uotienty the Index of whofe Logarithm is tc ; but - 
tc=t A-He — ta. Alfo if the Integer CD be divided 
by the FraSlim ab, becaufe a A=CE, therefore EF 
i^ the integral ^oiient whofe Logarithm is tE ; but 
tE=tA4-tC*^ta. Again, let th^FraBion ab be di* 
Vided by. the Integer CD, becaufe CA=ac, there* 
fore thq; FraSiion cd is the ^otient \ whofe Loga- 
rithm tc=:tA-|rta — ^tC \ that is, in every Cafe, the 
Logarithm ef the Divifor being taken from the Loga- 
rithm of the Dividend^ if to (he Remainder, yoa add 
the Logarithm of Uniity^ the Sum will be the Loga- 
rithm of the ^otient : which is but the Converfe of 
the RttUs for the Logarithms in Multiplication^ as is 
evident by Infpedtion of Art. 13. foregoing. And 
the Methods there mentioned for duely adjuiling the 
Indices^ are to be equally obferved here. 

. i8. In Involution^ or raifing the Powers oi Frac* 
tionsj 'tis evident that theDiJiance between t7»//y and 
the Root^ is equal to the Diftance between the Root. 
and thejf/y? Power ^ thtfirft gindfecond Power j the 
fecond find third Power^ and fo on. Therefore fince 
Aa=ab=ce=aeg^ 6fr. cdzz: Square ofab, efzizCube^ 
gh= Biquadratej &c. Power of the Root ab. But 
fince AB : ab :: ab : cd :: cd : ef :: ef : gh, (^c. there- 
fore the Logarithms t A+tc=2ta, and fo tc=:2ta — ^t A. 
That is, the Logarithm of the Square of the Root is 
equal to double the Logarithm of the Root lefs the- 

Logarithm of C/»/7y. Again, lit!? =atc«2ta— tA, 

and fo ta4-tc=4ta — 2tA ; that is, tex=:3ta — 2tA 5 
in Words, the Logarithm of the Cube is equal to 
triple the Logarithm of the Root^ lefs double the Lo- 

F garitboi 



34 TieNaturetf Log AJLiTHMs, and 

garithm oiUnUy. Moreover, ^^i^ t9=3ta — 2tAt 

therefore tc + tg=6ta — r4tA ; but tc=52ta — ^tA, 
fubduA this from the kfi EquatioHj there will re* 
main tg«s4ta — jta i or the Logarithm of the Bi- - 
piadrate. 

19. And uMiverfalfyj if the Pinver of a Frac^ 
tion be x^ and the Logarithm =: L9. the L <^a- 
rithm of the Power x (hall be esxL — ^*'-— i xtA, 
or ass ^ L— X t A 4- 1 A. Thus, fuppofe you would 
know the JLogarithm of the Square of the FraHim 
dc ^ here x=r2, and Lsstc, therefore 2tC— tA=tg» 
the Logarithm of the Square gh, as required. If the 
Logarithm of the Cube bedeftted of the FraSltontf^ 
we have x* ■= 3, Lrs te, and fo jtft— 2tA=tr, the 
Logarithm of the Cube rs, as deftPd. And thus you 
proceed for the Logarithms of other Pozvers. 

N. B. The Indices of Logarithms of all Paweri 
of Fr off tonal Numbers (l mean (iich as are furelf 
k>) muft be doubly dotted^ finice thofe Powers al- 
ways fall below the Roofj which is fuppoied a 
pure FraHion. 

20. Evolution^ or the ExtraBion of the Roots oS 
Powers^ is juft the reverfe of the foregoing Procefs. 
P or fuppofe^ the FraSiion cd were given, whofe fauare 
Root was required* Becaufe AB : ab rr ab :^, thete- 

fore AB x cdssab xab ^ and fo^ AB x cd=ab ; there- 
fore the Logarithm — "^-^^ =ta, is the Logarithm of 

ab tht fquare Root fought. Alfo if the Cube Root of 
th^Fratlion ef be required ; becaufe AB : ab :: cd : efj 
we have AB xef=ab xcd ; and fo the Logarithms 
tA + tc=s ta^-tc. But tA+tc— ta=5:tcs=2ta — ^tA, 

that is, 3ta=2tA — te, and confequently -^^--iS rrsta 
the Logarithm pf ab the Cube Root required. 

2fr 



Chap.IV. their ImUceSy farther expUtined* 35 

21. And univer&lly, if the Logarithm L of the 
Root of any Ptmer x, of any. Fra^on rs, be requiied* 
we have this«.<>rm tr+xtA-tA ^^ ti+f=mA 

=L=ste, that is, in Words, the Number* — ixtA 
added to the //ri^Ar of the Logarithm of the Fraiiian^ 
the Logarithm thus augmented being divided by x^ 
the Siuotient /hall be the Logarithm of the Root 
fought. Or, iincetAe=io, lOO, the Number at — i 
preftx'd to the Index of the Logarithm of the Power, 
and the Lo^rithm thus divided by x^^ the Sluotieni 
ihali be the Logarithm of the Root fought. 

C H A P. V. 

^tbe original ConJlruSHbn of Logarithms by the 
Lord t^cpcr^ and the Alteration thereof to the 
prefent Form by himjelf and Mr. Briggs, ex^ 
plairidandilluftrated by /i&^ Logarithmic 
Curve. 

I. riri H E noble Inventer of Z.e»/irr//£«r/, the Lord 
X Neper^ having duely contemplated their 
wonderful Nature, firft con(uruAed and publiihM a 
Canon thereof; but thofe far different from what we 
now comnKMily u(e. And this was no Wonder, fince 
Scarcely any thine receives its Invention^ and utmoft 
PerfeBion at the feme time. 

2. In the firft kind of Logitfithms that Neper pub- 
liflicd, the firft Term of the continual Proportionals^ 
was placed only fofar diftant from Unity/ tis that 
TVnw exceeded Unity. Thus, for Example, if o n 
be the firft Term ox the Series from Unity AB (fee 
Fig. I.) the Logarithm thereof, or the Diftance A n, 
or B y^ was by iiim put egual to vy or the Increment 

F 2 of 



5$ ConJlruStion ^Logarithms, 
jof the Number n y above Unity. If then we fuppoA 
yn=3H .000000 1, the Excefs of this Number above 

•AB=:r, iso.ooooooi, which therefore, by him, 
Yf^rmdtit&Logarilbm ; that is, An=0.ooooooi. 

3. From hence, by Computation, t}ie Number 
10 will be the 23025850th Term of the Series ; which 
Number thej-efore 1^ the Logarithmoi ip in this Form 
oi Logarithpts ; ^nd expreffis its Diiftancc from Unity 
ixi fuch Part whereof vy, orAn-i^one. Alfo the 
Logarithjn of 2 (in this Form) is 693 147 1 ; of 3, is 
10986122; of 4, is 13862943, {i?r. 

4. But this Pqfition of the Ratio of the Terms Is 
entirely at Pleafure \ for the Dijiance of the /r/? Term 
may have any given Ratio to its Excefs above Unity ; 
that is, A n may be indilfferently lefs^ equal to^ or 
greater than vy ; and according to that various Rath 
(which may be fuppofcd at pleafure) between A n, or 
By and vy, /. e. the Increment of thcfrft Term g- 
bove tlnity^ and the Drftance of tht (eimt from Unity ^ 
there will be produced different Forms of Loga- 
rithms, 

5. The Logarithms of this Jirji Form^ were found 
by thcfagacicus Inventer not to anfwer the Defign in 
the beft manner as could be wi(h*d ; and thereforb 
he changed *em into another more convenient Form^ 
wherein he putthe Number 10, not asthe23025850th 
Term of the Scries, but the locoooooth Term : And 
after Nepefs Death, the learned and indefatigable 
Mr. BriggSj with great pains^ made and publiih'd a 
Canon of Xiogarithms according to this new Form. 
Now fince in this Canon the Logarithm of 10 is 
1. 0000000, and fince i, 10, 100, 1000, £s?f. are 
Proportionals^ they fhall be equidijiant from each 
other ; wherefore ' the logarithm of 100 ihall be 
2.0090000 ; of 1000, 3.0000000, and the Loga- 
rithm of 10600 will be 4.0000000, and fo on. And 
this Form of Logarithms hath been ever fince in ufe, 
pd are thofe in pfefpnt IJfe ; the Nature and 

Properties 



Chap. V^ ipeplained and illujirated. 37 

Properties of which we have been hitherto explain- 
ing. 

6, The Rationale of the Method by which Mr. 
Briggs computed his Logarithms, is beft explained 
from the Lt^aritbtnetic Curve^ according to Dr. Keil^ 
as follows. In the Logaritbmetic Curve HBD (Fig. V.) 
let there be three Proportionals AB, ab^ qs^ very 
nearjy equal to one another \ that is, let their Diffe* 
rences have a very fmall Ratio to the laid Ordihaies^ 
(for fuch are thofe Proportionab,) and then tht Dif- 
ferences of the Logaritoms will be proportional to the 
Differences of the Ordinates ; that is, it will be 
sr : be :: Br : Be :: Ag : A<s, For fince the Ordinates 
AB, ab^ qsy are nearly equal to one another, they 
will be very nigh to one another ; and fo the Parts of 
the Curve Bj, B^, intercepted between them, will 
nearly coincide with z-Rigbt Line ; for it is poflibic 
that the Ordinates may be fo near to each other, that 
the Difference between the Part of the Curve and the 
Right Une fubtcnding it, may have to that Subpenfe^ 
2l Ratio lefs than any given Ratio, Conleouently the 
Triangles Bcb, Qrs, may be takea for Right-lined 
ones, and will ht equiangular : and therefore, fince 
a b is parallel to qs, they will htfimilar^ and their 
homologous Sides proportional j viz, rs : be :; Br : B^, 
or Kq : Aa. 

7. From hence, by the way, appears alfo the iS^^- 
fon of the Correction of Numbers and L^aritbms by 
Differences and proportional Parts. For putting 
AB=i, or Unity, 'tis evident, that the Logarithms 
Be, Br, are proportional to the Differences cb^ rj, ^of 
the Numbers AB, ab^ rs ; as we Ihall hereafter prove 
by Fa£ts^ in the praSical Part. 

8. If a mean Proportional ht found between I and 
ID, and then again another Mean between that and 
Unity ; and if proceeding thus, you continually find 
a mean Proportional between the Mean laji found and 
Unity ^ bifeifting the Logarithms ftill as you proceed 

(in 



38 QmfiruSiion ffViOQK'Bi i t h m s, 
(in the manner of the Ewmpk in Ch^p. 11. Art. 6.) 
you will at laft get a Number whofe Bijiance from 
Unity ftiall be Icfs than the tooooooo-ooooooTo Part of 
the Logarithm of 10. 

9. After Mr. Briggs^ in this manner, had made 
54 Extraaicns of the Square Root^ he arrived to the 
Number i. 00000 00000 00000 12781 9149320032- 
3442, and its Logarithm was 0.00000 0000000000- 
^555^ 1 1512 31257 82702. Suppofe this Logarithm 
be equal to Aj or Br •, and let the Number found by 
thbExtradion, be=:;i; and then its Excefs above U- 
nity will bc=rj. 

fAj=aO.OOOOO 0000006000 0555 1 11512- 
1 3125782702. 

ThatiSy^ ji =1.0000000000 00000 1 2 78 1 91493- 
20032 3442. 
rj=so.oooooooooaooooo 12781 91493- 
200323442. 
low by means of tbefe Numbers the Loga- 
rithms of all other Numbers maybe found in the fol- 
lowing Manner. Between tht given Number (whofe 
Logarithm is to be found) and IJnity^ find (by the 
ExtraAion of Roots, as above) fo many mean Pr9^ 
fwrtumals till atlaft a Number be obtain*d7& little ex- 
eeedingUnity^ that there be 15 Cyphers next after it, 
and as mznyjignificant Figures after tboie. Suppofe 
the fmall Number thus found h^ab^ and let the^^- 
nificant Figures with the 15 Cyphers prefixed, before 
them, denote the DiiFcrence be ; then fay, as the 
Difference rs is to the Difference bcy fo is the given 
L$gariibm Br, to Be the Logarithm fought for the 
Number a b. If now tRis Logarithm B^ or Aa^ be 
coniinually doubled the fame niimber of times ^ there 
yreft Extra&ions of the Square Root^ you'll have at 
]af( the Logarithm of the Number propos'd as re- 
quired. 

II. If the Tangent TB be drawn to touch the 
' Curve in B, thcii may the Subtangent AT be found 

by 



10. N 



Chap. V. explained and ilbijirated. 3^ 

by: the Numbers above in jlrt..^. For fince AB, 
Br, are parallel torj, AT -, therefore the RigU4hfd 
triangles Bri and BAT, are fimilar^ and fo as 
sr : rB :: AB ; AT, theSubtangent j but fince ABsi, 

therefore ^=AT ; or thus 5 

sr 

As rx =0.00000 00000 00000 1 2 78 1 9 149} 10032- 

Is to Brzso.ooooooooQO 000000555 1 11512 3x257- 
8270, 

^ Is AB=::1 .00000 00000 00000 00000 00000 OOQOO 

OOOO, 

to AT=o.434i9448 190325182^65 1 1289 18915. 
. 6051. 

12. If the proportional Right Jams GH, EF, AB» 
CD, (Fig. V.) aie Ordinate to the Axis CV of the 
Logaritbmetic Curve^ and if their Ends FH, DB, be 
joined by Right {Jnes^ which produced meet the 
Jxfs in the Points P and K, then the Rieht Lines* 
GP» KA, will be always equal. For fince GH : EF 
::AB:CDi itwiUbe, asGH:Ft:: AB : DR (by 
Chap. II. Art. 8.) But becaufe of the Jimilar Tri- 
angles PGH HtF, as alfo KAB BRD, we have 
PG : Ht (:: GH : Ft :: AB : DR) aKA : BR. 
But HtsBR, and therefore PGsAK ; which was 
to be denHMt^rated. 

13. If the Right Lines CD, EF epially accede to 
AB, GH, fo that the Pbint D at laft may coincide 
with B, and the P<Mnt F with H, then the Rights 
Lines DBK, FHP» which did hfore cut the Curve, 
will now only touch it in the Points B and H \ that 
is, they will be changM into the Tangents BT, and 

' HV. And fothc Right-Lines AT, GV, wiUtfA 
ways be equal to each other ; and fo the Suhtangent 
AT, oa GV, in whatever Part of the Axis it be, is 
always me cmjlant given Length ; and this is one of 
the moft remarkable and ufeful Properties of theZjGK 

garithmic 



40 A Method of conjlruSiifi^ 

garitbmic Curve : For the different Species 6r Formi 
of thofe Curves are deter m i hed by the SubtaHgents. 

14. If the Excefs cb of any Number a b ejttremely 
near Unity, or but a fmall matter txceedihg \t\ be 
given, the Logarithm of its Diftance from Unity Aa 
or Br, will be known by means of the conftant Sub^ 
tangent AT ; for by Art. 1 1. we have be : Be :i Afi 
: AT ; therefore ATx^f=Bcx AB=:Bc, the Loga- 
rithm required. Thus alfo AT x rj=Br, the Lo^- 
rithm of the Number qr ; and fo the Logarithm of 
any prime Number 2, 3, 7, 11, 13, 6fr. may be 
found independently of the Logarithm of any other 
Number.. 



^"^n^ 



C H A p. VL 

ji Method of conJiruSiing the Logarithms derived 
arni detnonftratedfrom the Nature of Numbers 
only^ by Dr. Edm. Hallcy, 

I. 'T^ H E admirable Method now before us, is 
X one of the many great and wonderful In-' 
ventions and Difcoveries of the celebrated Dr. Halley^ 
the prcfent Afironomer Royal^ and Fellow of the Royal 
Society*^ and whofe Name amongft the Literati will 
be had in ever lofting Remembrance. This Method 
not only comprehends all the Improvements made 
by others by means of the Hyperbola and other Geo- 
metrical Figures^ but fhews with great Accuracy 
from the. pure Properties of Numbers (as moft natu- 
ral and agreeable in a Bujinefs purely Arithmetical^ as 
the Logarithmotecbny is) bow the Lc^rithms may 
be produced to any dejired Number of Places^ with 
^r greater Eafe and Expedition than by any Method 
before known. According to hio^ therefore^ 

2. 



Chap.VL 5/*LoGARiTHM8. 41 

. 2. Suppofe an infinite Number of equal Ratios or 
RatiuncuU between any two Terms in a continued 
Seale or &eries of Proportionals ; and thofc Ratiun^ 
cula exprefs the Ratio of thpfe two Terms, as of 
I to i-Hx. If then between Unity ( i ) and any Num^ 
her propofed (i-4-Ar) there be taken any Infinity {n)o( 
' mean Proportionals^ the ihfinitely Jittle Augment or 
Decrement of th^firft of tbefe Means from Unity will 
be a Ratiuncula or Fluxion (*) of the Ratio of C/zri/y 
to the iaid Number; and the Terms of the- Series 

will ftand thus ; viz. i. 1+*^. 1-+-^. i-¥x, i-hx 

&c, to i-hx . and the' 

Exponents} ^- ^' ^: 3; 4; g^- to «; or thus. 

From whence *tis evident that not only the Number 
C»j of the Proportionals or Ratiuncula^ but alfo their 
Sum (nx) may be put for the Logarithm of i-f-^. 
And thus alfoJV^ maybe put = Logarithm of i-hy, 
and confequently it will be, as L, i+x : L, I-+7 
:: nx : W, 

3. If;e=i, that IS, if the RatiuncuU compoiing 
divers Ratio's have the fame Magnitude, f hen are 
thofe Ratio's proportional to the Numbers of Ra^ 
tiuncuU cbntain*d between their Terms, viz. 
L, 1+^ : L, i^ :: n : JV, For if Af=i, 2Lndy=2 \ 
then i-4-^fc=2, and i-+y=3 ; and if »=3p|03, t?r. 
iVwill be found =47712, fcff. that is, if there be 
the infinite Number 30103, &?f. of RatiuncuU in the 
Ratio of I to 2 ; there fhall be the infinite Number 
/^yyii^ fcfr. oH)\Q hm^ Ratiunci4^ \vit\i^ Ratio oi 
I to 3. 

4. On the contrary, \f nis^N^ then L, i-^rx j L, 
i+y :: x :y^ that is, fuppofe the infinite Number qf 
Ratiunculae in one Ratio equal to the infinite Number 
of RatiuncuU in any other Ratio^ then arc the Lo- 
garithms of thofe Ratio's diredlly as their Fluxions^ 

G * or 



42 A Method of conjirulfif^ 

or as the ^Magnitudes of the RaUuncuU refpedively. 
For'inftance, let the Ratio of lo to |, jog to i, 
*iooo to I, i^c. all and every of them be fuppofed 
to confift of 5 Rat\unculae i as follows; Thus, 

{Ratio's o. I, a. 3. 4. 5. &ff» 

Terms i. 10. 100. lOOo. loooo. looooo, &ff. 
Ratiunc. 51;. 5w. 5X. 57. 52, i^c. 
*Ti3 plain the Ratiuncula v of the whole Ratio of 10 
to I IS T 5 of 100 to I, is I, f5?r. that is, the Ra-^ 
tiuneuU are/i;=rj, w=t, ^rs^, j^=:t, 2rrii butthofe 
Fr anions are as the natural Numbers i, 2, 3, 4, 5. 
Wherefore the Logarithms of the Ratio's of lo to i, 
100 to I, 1000 to I, Cf?f. are' diredly as the natu- 
ral Nunibers I J a, 3, 4, 5, fi?*r..and fo may be ex- 
preffed by them. 

5. Since then the Logarithms of Ratio's are as 
their Fluxions^ therefore the Logarithm of any Ntim* ^ 
ber is found by taking the Difference of Unity and ^he 
infinite Root of that Number ; that is, becaufe 1+^ 

is the firft Term from Unity, ot Ratiuncula j i-^^" 
:;=i4-Ar, is thei infinite Po^er to be refolv'd \ and 

•7+3, or i+J''=?=|-4-ir, andfo i-f-^^" — i=3c^ Lo- 
garithm of ^-+-Jif. 

6. In order to extraft the Root of the infinite Power 
iH-x, (which, to fome, may fcem ftrange and next 
to impoffible) we muft make ufe of Sr Jfaa,c Newtqn% 
celebrated Theorem for that purpofe* Suppofing 
then the Power be iH-at, according to his' Theo? 

rem, I -f- *"==! I + J^AP 4- -^JIT- ^Af + — 6^3 — * 
,-6»4.,„»_6»» ^^^ ^^ ^^ ^^ ^f^^ p^^gp 

i-^x^ when the Index (n) is finite ; but (n) being in 

. the prefent Cafe infinite^ all the Terms of the Co* 

efficients^ wherein (nn) is found a.Divifor (as being 

inf^nitely in^uite) will vftniftij^ as being infinitely le^ 



chap. VI. e/'LoOAkiTHMs. 43. 

than nothing. But the Co-egicient ~ = i-— 1 =- 

Coefficient '-6>.+..;»-6>.f = ±^_ ± =: Ij &c. 

Wherefore the foregoing Root will become i-^-i^ — 

inx'''+ \nx^ — inx^-^-inx^ &c. That is^ 

t 

ixx-^^lxx-hixxff — i*^ — ix% &c. =i+^* — l=si>=: 
Logarithm of i '^x. 

y. And whereas the infinite Index (») may betaken 
at pleafure, ah Infinity of different Scales of Loga- 
rithma may be produced ; and thofe different Loga* 
rithms will be to one another as j^, or reciprocally as 
the Indices (»). And as it hath been (hewn (in 
.Chap, v.) that in making thejfr^ kind of Logarithms 
by Neper ^ the infinite Index of the Logarithm of lo 
would be 23025850, &?f. But in making the fe- 
cond fort after by BriggSj the faid infinite Index was 
put =10000000, Csfr. Confecjuently in the fore- 
going Series for Logarithms, if «=iooooooo, dfc. 
the Lord Neper^s Logarithms will be produced i and 
the Scries will hejimpfy x^^x^^ix^—ix^J^ix*, &c. 
On the contrary, if »=:23025850, &fr. thenBriggs^s- 
Logarithms will arife from the Series j and bccaufe 
» = >.3oa,8o f £sf^. =0.434229448, i^c. therefore 
J(=:AT) is the Subtangent of the Logarithmic Curve 
for the Briggian Logarithms^ as is plain from Chap. I. 
V. Art. II. Whence if a Logarithm of Neper*s 
Form be multiplied by 0.43429448, 6f^. or divided 
by 2.3025850, ^c. it is converted into a Logarithm 
or Briggs's Sort^ or thofe in prefent Ufe. 

8. If the Logarithm of a dccreafing Ratio be 
fought, as of I toi— X, the Power being i — x^ its 

■L 

infinite Root will be i— x*=i— i;^— *»^*— i»*'— 

G 2 iwx-*— 



44 A Method of conJlruBing 

inx^—inx\ &c. that is, i x x-f ix^-H^^+ir^+TJf *» 

&c. =1 — I — :e^z=Sif=i the Logarithm of i — x j the 
firft "Term next below Unity, or Root of this infinite 
dccrcafing Series being i— ;r. And fo in this Cafe, 
according as the infinitelndex {n)is made =10000000, 
6fr, or 2.3025850, 6?r. fo Nepet^s or Briggfs Loga- 
rithmsof thofe negative Numbers will be produced. 

9. Inftead of the Terms i : 1+;^, let ^ : * expreis 
the Terms of any Ratio univeriaily 5 and make 
^+^=^» ^^d a — b=::d ; and fince it is i : i+x 11 aih^ 
therefore /i+^*!=i, andfox= tzi — L . Again 

becaufc (in the decreafing Ratio) it is i : i— jc :: ^ : tf ; 
therefore b—bx=a^ and fo we have again x = t^ 

= 7 • Whence the Logarithm of the feme Ratio 

a : b^ may be doubly exprefs'd ; viz. for the encreafmg 
Ratioj the Series will be 

the decreafing Ratio 5 all which is evident from the 
. three laft Articles. 

10. But if we fuppofe the Ratio of a to b^ viz. -^ 

compofed of two Parts ; viz. of the Ratio of a to 
the Arithmetical Mean between the Terms a ^nd ^, 
and of the Ratio of the faid Arithmetical Mean to 

the other Term by that is, fuppofe ~ =-^ v ^ 

(for is = tt^ is the Arithmetical Mean between a 
2 

and b'^) then the Sum of the Logarithms of thofe two 
Ratio's, ~> ^, will be equal to the Logarithm of 

^e Ratio of a to b. Or, L, ^ + L, ^ =L, ^ . 

And 



Chap. VI. g/'LpGARITHMS/ 45 

And thus alfo we have L, ii+Li^Li 
Now bccaufe the Ratio of 4i to * is encreafing, thctc^ 
fore I :i + x::is:l^i and fo i -f ^= i; con- 
fequentlyx= •; i= -y-^ = — a-new; again, 

becaufe the Ratio of is to a is decreajing ; therefore 
4j : ^ :: i : i — x, and fo i — ;^ aes J! ; and 

1 1. Therefore fince ;f = i for both Ratio's, vi%. 
of tf to ij, and *j to ^, we (hall have (by the fore- 
going Rule, Art. 9.) Jx 1 4. 4 4.-I!! 4,^ 

4--^. 6?r. =(A)L,tfto4j, andjx i_j: 

+ 3? ~;^*+P'^^- = (»)L.4,to*. Then 

.-'' 7 • + ^ • + ~, .t?^. = (A+B)L,ato*. 

Thus you have a Series exprcfling the Logarithm of 
the Ratio of ^ to b^ whofc Sum is szzaJ^b^ and Dif- 
ference d=<j4"^' and this Series convenes twice as 
fwift as the former in Art. 8, and therefore is more 
proper for making or ^flf^w/W^g- of Logarithms, which 
it performs with great Expedition, when d the Dif- 
ference is but the looth Part of s the Sum ; the firft 

Step — fufficing for 7 Places of the Logarithm, and 
thefecond -7 for 12 Places. 

12. Becaufe the Difference of the Logarithms of 
the Rations of ^ to is, and is .to ^ is the Logarithm 
of the Ratio of a ^ to |ss ; or thus, becaufe 



^) 



46 A Method of confiruSting 

(A-B) = J X ^'^. -f ^ ,^4.. 1^- , ' 6fc. but 
half the Ratio .-is the Ratio ^ (for J^ x ll? - 
^), that is, the Ratio of the Geometrical Mean 
y^l^b to the Arithmetical Mean i s j confequently 
theLogarithmofV:g= J x i! . ^^t . -f Ji;, 

&? f , which IS a Theorem of good Difpatch for find- 
ing the Logarithm of is. 

1 3* But the fame Logarithm may yet be much more 
advantageoufly obtained by a Method like the for- 
mer. For if we make the Terms of the Ratio 

^ =-^, and put 5 = tfi 4. iss, the Sum of the 

Terms , and D = ^ ^ — iss ; 'tis evident the 

i2l » f2!, ^c. by Art. 11. But becaufe iss =' 

ia^ + iab -f- i^*f therefore D = tf i — iss s: tf ^ — 

id" -{-iab^.ibb =« 4^* — iab + i^^ = v^l^^^TIJ 
= id* =1, in the prefent Cafe of finding the Loga- 
rithms of Prime Numbers ; for fuppofe the Loga- 
rithm of 23 be fought, then ^==22, ^= 24, issziaj, 
and d 3= 2 ; alfo -3^=^^ = 528, and fl=isss=s529, 

and therefore ab — hs;=:A—Bz=Dz=i(^:zll. Where- 

• 
fore, finceD=i, the Series above becomes ^y, — j. 

■iS5 + j-f^ + tT^' ^'- ='^^ Logarithm of the 
Ratio of -r* i a»d fo the half of it, viz. ' the Series 

i '^ i + pi + iH-s + ?5^' ^^- ■ =*^« Loga. 

richm 



Chap. VI. g/'LoGARiTHMs. 47 

rithm of the Ratio oi^ab to is. • And this converges 
much fooner than any Theorem hitherto invented, 
and beyond which nothing better can be hoped for, 
in tht great Author's Opinion. 

14. The Logarithm given to find what Ratio it 
expreffes, is a Problem lblv*d with like Eafe^ and de- 
xnonftrated by a like Procefs to that foregoing for 
finding the Logarithm of ^ given Ratio. For as the 
Logarithm of the Ratio of i to i-j-^ was proved to 

be i^" — I , (by Art. 5.) and that of the Ratio of 

I to I— ^ to be I — I — x» rby Art. 8.) fo the Loga- 
rithm, which we will call L, being given, fincc 

L=i+;tf» — I, therefore L+i=i4-^ in the firft 

Gafe; and i — L=i — »vw, in the latter: Confe- 



quently i+L =i4-^> fl'id i — L =1 — x. That is, 
according to Sir Ifaac^s Theorem^ l-|-»L+»^*L.*+ 
J»«L«-f ^♦L^+TT^w'LS £s?c. r=i-|-;ri and alfo 

I— 11L4- 4ii*L*— . iri^U-ir ^n^l-^— ^izn^L^ &?r. 
x=i — X ; confequently the Number i+x or i — x 
is readily known by thofe Series, be the Species of 
the Logarithm what it will. That is, whether it be 
iVif^^r's Logarithm, where »=t=iooooo, &c. and fd 
l±xz^l±L+iU±lU+^\L^±^i^L^, i^c. orwhe, 
tlier »=2 3025850, &c. for Briggs^s Logarithm. 

15. If one Term of the Ratio^ whereof L is the 
Lx)garithm, be given, the other Term will eafily be 
had. by the fame Rule. Let a = the leaji Term of 
the RatiOy and b = the greatefk •, then becaufe it is 

I : i-i-xiia :ti and foi-f-xss — =ri^L + 

. iL* + IL* + T^L^ ^c. if»=ioopooo, &c. and 

therefore b=s: a +'tf L + i<^L* + iah\ &c. if a were 

|iyen j but if ^ were given, becaufe i ; i—x :: h a^ 

therefprQ 



4$ A MetbodofcorjftruBing 

therefore i — y = 4 » *^d ^^ a^=zb — ^L + 

i^L— i^L*+ AL, esfr. Wherefore by the Help 
of the Tables^ the Number belonging to any Loga- 
rithm will htexaSly bad to the utmoft Extent of 
the Tables. 

1 6. Suppofc — = -A^f the Number belonging to 
the given Logarithm L of the Ratio — , then 

1 == tf, let the Logarithm of the R^tio ^ be 3, 

and let the Term b b^ known ; then (per Rule, 
Art. 15.) we have i : i—x :: b : JNT, and fo i — xi == 

i?, andiVs:* — ^ + i*aa— l^asfcf^. if a be 

Nepey*sLog2LTithmyhutN=.b — bn'^+ibn^'i'B — ln*h^y 
6fr. if 1; =5=2. 3025, &c. as in Briggs^s Logarithm. 
But if the Ratio be 4 == JV, then ^^ =; *, and fo 

I + ^ = ^ , therefore iVsatf+tfa+itfaa+I^S 6?f . 

Or N.= a^an'3i+ ian'"h*-^an^'^\ iSc. Note, here 
a and ^ denote the Number belonging to the neareft 
ntxt lejfer ornext greater Logarithm than the^/i;^» 
Logarithm L, and the Logarithm g is the Differenee 
of thofe Logarithms ; wherefore as a is I'efs, the Se- 
ries converges thefwifter ; and finds the Number N 
of the Logarithm L, much, iboner and eafier than 
the Rule in Art. 15. 

1 7. In the foregoing Series tf+^H-f i^aa+i^S^?^. 
x=N\ the three firft Steps may be abridged thus, 

a -j J a = ^ + ^a -|^ 1^33, very nearly, and may 

ferve with ExdSnefs enough for Numbers not 
exceeding 14 Places, which is more than fuffi- 
cient for common Ufe. Therefore we may take 

a^ 
3 



1 



trhap. Vi. tf t od AHt T Hli«; 41 

a+ l^^N, or3 — ^=:i\rjand if the 
Jfei/«r » taken for Bri^h Logarlthihs, wc A^ have 
^^T^^^ N,ov b^ l£^.=^Ni that is 

**rl^=i iNTj which Ecjuation may be rcfoIv*d jnto 
the following Analogies ; 



w;^. i ^~!^ • "^"^3 •• ? i ^5 Ofi 



€8. If more Step^ pf /i^f 8mu be defir^d, it wU) 
be found as foUows, i^/x ^^^ fJL- — "dfi* -x, 

^i^ ,&c.=!^Ari alfb theiRule t-fwL+iMLL^*' 
i»^L', 6ff. may be thus contracted, Hz. i -^^ 

2Hh»I>H»»LL xi^Lss JV. What is laid concern - 
ing tKs Method 'of making, ^rither LigaHtbms or 
Numb^s^ J ()fic6iGi^;ls fuftcient to ^dcrit very in- 
telligible to any common Capacity; and to ihew the 
admirable UJefulntfs and Exc^lkniy thciwf bi^Md 
iny other^ Mtherto ifivented. 



1^ r CHAP; 



]^d^ 72^ Logarithmic Series 



CHAR vn. 

1t^ * Logarithmic Series aforegoing^ demons 
firated aljb ^ *Fluxions, amfroni the Nth^ 
' ture of the ♦Hyperbola. 

x.rr^HE preceding Series for the Logarithms, 
X ^bich has, been demonftrated furelj from 
jdrithmeticd Principles^ or the Properties of NuM" 
bers^ may a!(b be prov'd from the Do6bine of the 
fluxions of L^ariibms. For the Writers on Fiux* 
ions varioudy aemonftrate she Fluxion of the Loga- 
rithm ofas^ Number is equal to the Fluxion of that 
Number, (wbofe Logarithm it is) divided by tbefaii 
Number it f^. 

2. Let the Number propofed be i+^, the Flux- 
ion of which is jr, therefore the Fluxion of its Loga- 
rithm will be ss -~- 5 from whence the foregoing 

Jnfioite Series for the Logarithm of the Number i+^ 
may be derivedi as fqllows. The Logarithm <^ 
l-^x is equal to the flowing Quantity or Fluent of 

the faid Fluxion -i^. But ^r"^ ^ x -j- • and 

Jj^vszl+X) I (=1— X-^2J*P— XS &C, 

i-^x 



• — X 
— X — XX 




'+XX 
'I XX \ x^ 


» 




-X* 



-tVf*, &c. 



chap. VII. JeHmftratedby Fluxions. $t 

3. The Quotient then i—H-x*- ■ ft* I » S Uc. 
1_ J but *x 1— *4-x*— «»+x*, &c SB i~ 

j^jj4.^,ji«_,c»i-f-J^*,&c«— -^. The i7««i/ there- 

fore of that infinite fiuxienary Sam (by the inverfe 
Method of Fluxions) is found to be *L_i^^4^_ 
»^_i_|x', &c which therefore is the Logarithm of 
dje Number l+* ; and is the very fame with that 
Scries in the foregoing Chap. VI. Article 7. for Ne- 
pet*s tfigoritbms. 

4, Again, if the Ratio be decreqfing, or the 
Number be 1—*, the Fluxion of this alto is*, and 

therefore the Fluxion of its Logarithm ^. But 

i__ « ir X -i- as lH-;f+X»+*», &C.X * ss A + 
l—x «— * 

xi-f ^*+x»x, &c. The Fluent of which is *4.fe(»4. 
7X»4-i**, &c. the fame Series as that in Article 8. of 
the preceding Chapter^ for Neper's Log^uithm of 

5. The fame Series is likcwifc deduced from the 
Nature or Equation of the BfperMa. For letFCH 
be an Hyperbola (Fie. VI,) AE, Al, the Afymf totes 3 
draw BC, DC parJlel to AI and AE j alio draw the 
Ordinate EF parallel to the Ordinate BC, or Afymp- 
tote AL Let AB=«, EF==y, and BEsswf. The 
Equation afl=tfy-f ;gf expreffes the Nature oftheiify- 
perbola between tU Afymptotes. Now the Fluxion 
of the Space between tht Jifiiffaf Ordinate^ and 
Curve, is always equal to the Proek^ Qi the Ordi.: 
nate into the Flumon of the Abfciffa j that is, in this 
Cafe, s»ri. Therefore to determine the Fluxion of 
the Afymptotic Space contajnM between the Abfcifla 
BE, the Ordinatea EF, EC, and the Curve of the 

HjrperbolaFC, we have ^-^^s J', thcK^fore^ as 

f *= Fluxion of the (aid Space FCBE. 

H 2 6. 



'^4 *thi ZAgariibrntf Smtt, Bcq>' 

6. But ^ =r ^ xi, and ^ = a-f*) a0 
(=«— *-f.Sf_i!H- '4, &c? Wherefore ^ = 

MTf ^antity of this fluxionary Series is ax — : be* -¥■ 

fa-T^-^^^ *^' - ^^ Space FCfiE. Sup. 
pofe^j=i, then x— |yM-4^^— i^^-Hx% &c. = tbc 
{kidyij^fnpiofic Space FCBE, as before, But 'tis cvi- 
4ent this is again the very Tame Series as was invented 
by Pr. Halley for the Logarithm of the Number i4^. 

7. The ^ymptotie Spaces^ then, are with rcfpcft 
to the Abfcijfie^ as Logarithms in refp^ to Numbers. 
Thjit is, Cnce AB=i, and the Logarithm of 1 is=o^ 
the Spaces Bu^C, BfiC, B^^G, Bi&)fcC, 6?r. 
are the Logarithms of the Numbers A tf, A r, A ^, 
A i5, &c. Again, be^aufe the Jbfcijfie are in a r^- 
tiprocal Proportion of the Ordinates^ that is, 
AB : AE :: EF r ,BC •, thereforethe Afymptotic Spaces 
are in refpeft of the Ordinates as Logarithms in re- 
foeA of Numbers: Yet fo, that while the Ordinates 
BC, ab^ cd^ eg^ hk^ EF, iecreafe in a Geometrical 
Rirtio.tht S^caBabCj BcdC^ 6?r. may in- 
cfcafe in an Arithmetical Ratio. And fince in Ncr 
per*s Logarithms J x x --^ixx'+^^xx-^ix^y &c, 

praiopooQt &Cp *tis plain his Logarithms become 
the fame witK the Hyperbolic Logarithms juft now 
^pnfidcr'4. ' ' 



Sf 



CHAP, 



Chap. V|II. OfhoGAKnnMSy&c. 13 



C H A P. VIII. 

fTbe Method of conjlru0ing Logarithms by the 
Infinite Series^ exemplified and iUuJlrated. 

l.fTPiHE Manner of raifing Theorems for the 
X Conftrudtion of Lc^rkhms hath been luf- 
ficiently explained \ it theretore remains that we il- 
iuftrate the fame by proper Examples. TkeTheo^ 
rems for doing this direftly are. 

Theorem I. J x at =p *;? *-4- 4x'=p ixH- \x\ &c. 

-s L, izbv. 

Thtoftmir. J X f 4- ;-^ + ^^ + ^\ 6?f. 

— T * 

Theoremlll. i x ^i + ^ ,-+- ~ 4- ~, 

Theorem IV. Jx|- 4- jgj -*-j-p -♦- -g,- 

&c. = L 

.Note, in thefe Theorems, J is all along applied to 
^apt them to dl forts of Logarithms. 

2. Since »=the Logarithm of lo, we muft there- 
fore firft fupppfc »=iooooooo, &?f. and thence Ne. 
per's Logarithmi will be produced j and fo thefe are 
the firft fort of Logarithms which Nature affords: 
TheothwB, as Brigg^s Logarithms, &c. are made 
from them. In order then to find a Briggian Loga- 
lithip, 'tis neceflaiy firft to find Utptrh Lo^rithn\ 
of iQ. • This may be done feveral Ways, either by 
the Number lO itfelf, or by ife component Parts. If 
we attempt it by the Number ip it felf, then becaufe 




54 Of conJlruSUng Logarithms 

i-hte:io, we ihall have x=:9, which, becaufe it is 
greater than i, will occaiion that the fir ft Theorem 
will not converge % and fo the fecond Theorem muft 

be ufed. In this t-=tv> and therefore i= i + lo 
ss 1 1, and ^io--'i=9. And thus the fecond Se- 
ries for Neper's Logarithm of lo will be ^ H- 

2.30258, &c. the Logarithm fought. 

3. But this Series converging fo eiHremely Jlow^ 
renders the Bufinefs very tedious, and therefore the 
faid Logarithm muft be attempted from the com- 
fmtent Parts of lo. And fince 8x ii=:io, and 
2 X 2 X 2 =?: 8, therefore 3L, 2 H- L, li = L, 10. 
Confequently by finding the Logarithm of 2 and li, 
we find the Logarithm of 10. Now Neper^s Loga- 
rithm of 2, is found either by Theorem I. which con- 
verges very flowly \ or by Theorem II. which con- 
verges much fafter; and therefore to be chofen. 

Here ^ t=z }.^ and /H-*=^=3, a — b=d^i^ and fo 
the Theorem £? 4- fl' 4- 5^ , .fc?r. = ^i -t^ 

&c. = L, 2 ; this multiplied by 3, is = -^ -j- 

3 

ix ^ + ixf7.,£5?r. = 3L, 2. Nowput^,=A, 
andbccaufc ^-r x p i= p , and' ^x = p therer 
fore alfo put f A -B, and foB^z t. •, and thus 

|B = C, 4C = D, and foon. Whence the The- 
orem will become 2 -tr ? A + i BH- 1 C +| E) -H 



Chap, VIIL hy infiniti Series. 55 

4. By the fame Theorem IL we obtam Nefeft 

Logarithm of li; for becaufe -i = li, therefore 

L «,iy,(^i. = f.. Whicea+3=.i=9, 
and d = tf '^^ =? I ; and fo the Theotem ^-f- 



^%, ^c. =|^-txL + ixJJ. e?r. 



2d3 

But becaufe I 



= iD=sE, (^c 



therefore the (mcI Theorem will become AH- iC+ 
iEH- fG4- SI-I- T^L, 6ff. =L, 1 i as required. If 
now this Series be added to the foregoing (m Art. 3.) 
wc fhall have the Theorem 2+ itA+ 18+ ilC-j, 
tt)+ ifE+ tVF+V1^4. %VH, fcff. = jL, 2+L, 
]4=L, 10. See the Operation in the Table below. ' • 



I. 

I A=;0.22222222222Z2 
B =: . . 246913580246 

C = . . . 27434842249 
D=. ...3048315805 
E = .... .338701756 

•'? = 37633529 

■M.== 4181503 

H= ....... 464612 

I =;: . 51624 

K=. ...:... .5763 

L= 637 

M= 71 



2. : 
llA: 
tB- 

*»C = 

iD=s 

I^E = 

tVF = 

tVHs:. 

18 T 



: 2.ooooooooooooe 

:. .29629629629^2 
:. . . 49382716049 
:.. . 13064210595 

: 33870»75^ 

; 9^P4io 

: 2894887 

876124 

I... .2733Q 

■- 8454 

273 

85 



Thus Neper's Log. of io=2,3025850929940,&ff. 



5. The Logarithm thus found (if continued on) will 
be 2.30258509299404568401799 14546843642076 
01101488628772976033328, &r. =»» and there- 
fore 



56 Of am/iruSting Logarithms^ 

fore in making the Briggian Logarithms^ we iKall 
liave j^=o.43429448i90325i82765ii2'89i89i66o 
50822943970058036665661 14454, &c. the Reci- 
procal of the former ; which hencefcnth let be call'd 
% ; that is, let J^. And now we are prepared to 
iBnd the Logarithm (of any Form) of any oAcr 
)«fiimber. 

6. For Example, let it be required to find J?r/jjyA 
X^Karithm of z, to 10 Places of Figures. In order 
to mis, the Index J muft be aflTumM of a Figure or 
pm more than the intended Number of Places ki the 
Lo^rithm. The fecand Theorem is moft proper for 

this Purpofei for here again i = f , is=: i, and 
J S3 3, and alfo zso.434294481903 ; and the 
Theorem Jxi? + !^ + ?!!, I6c. =zx| + 

'}x^ + 1 — , ^c. =L, 2, and therefore -^ -f* 
txp +ixp,&c. = JL,2. Hetcput^=:At 
and becanie ^ s — x ^ s:txA=:B, and thvs 

*. s«— X ^ =fBs=BC, andfoon. Therefore 
S' 5^ 3* 

the (aid Theorem will here again become A4- f B+ 
fC-f- il>-hiE-l- iSF, £sfr. = tL, 2, as is evident 

:&om the following Operation. 



I«d» 



Chap. Via, fy hifimte Seriet, 



$7 



\z =UV. =0.144764827301 


A— 0.144764827301 


B= 1 60849808 1 1 


»B — 5361660270 


, C=s 1787220090 


K= 357444018 


D=s 1985800I0 


ID- 28368572 


Esa 22064445 


jEsas 245*605 


F = 2451605 


t^F= 222873 


G= 272400 


AG=s 209^3 


H=s 302^6 


.Miss 2017 


I = 3362 


•M= m 


K« Z7i 


.-^Kaa 19 


L=: 41I 


AL=s 1 


The Sum is *L, 2=0.150514997826 


Moldi^edby 2 


The Briggian Logarithm of 2=0.3010299956521 



7. This Logarithm may yet be much eaficr and 
fooner obtain^ by this Confideratioti, viz. That 
1""= i^» and -JIHxi^^ =r5b* therefore 

L ioee I T \ T « 

r5i4^^^Tg7o ^s r* «o»t =3 L 4 = L 2. But 



10 



10 



L||ff=:LiH* Whereforeput J|i=-J a-new, then 
4i+i==^sss253, and^h— *s»i=3i thusTheofemll. 
will convergsB much After, and will become, in 

Numbers, z xTxTfl 



fx 



2533 



«$3 



Or if — =jr, the fiid Theorem, in Species^ h 

j^ _f. |2jy» -f- :^zy, &c. Snppofe 225^ s= A, then 
|z^»=Jx2ayxy*=3iAxy*=B, alio ^iy'=r|x2zyxy»xy»=s 
iftcy*=C, and fo onj and thus the Theorem is 
A-KAj'(=B)-MBj*(=:C>HQV(=D)+iiy,&c. 



8. 



58 OfconftriiSlingLoGAKiTKMS 

{2zy=A= 0.010299473879 1 2 
tAx5!X=B= . . 48271995 

iBxjy =C=: . 4P7Z 

TheSum is the Logarithm I ^^v^«^^^^/:/:«^q^ 

^f J_06o/ T 115X^1 *.. 0.01029995063980 

Add the Logarithm of ~; . . . 3 .oooooooooooooo 

And A of that Sum will 1 /:/• o 

beL2== I ..0,30102999566398.0 

Thus you fee 3 Steps of the Series thus ordered, are 
fufficient for 14 Pkccs of Figures, whereas before 
(Art. 6.) 1 1 Steps produced the Logarithm true only 
to 10 Places. 

9. Let thp next Example be to find the Briggian 
Logarithm of 3. This may alfo be don6 by Theo- 
rem II. where J^ = f-, and J=2, and j=4; alfo 

~ rr i, 2=0,43429, &c. as before. Then 2 x ^ + 
3S» T js' i 8 ' 32' '"• 

= L3. The half thereof^ +7X^ +^x * 
&c.= i L 3. Put -5 S3 A, then 4 = ^ X i = 

Z o Z 

^A = B, and ^^ = -^ xit:^i^ = C, andfoon. 

Whence A 4. }B+ iC+ tD+ iE, &?r. =iL 3. 
But this Series converges fq very flow, that as many 
Steps will be neceflary as you intend Places of Fi^ 
gures in the Logarithm, and more ; therefore Re- 
courfe muft be had to Theorem IV. which here comes 
into play, becaufe the Logaritkms oa each Side of it 
are known, viz. the Logarithm of 2 and 4. 

10. Therefore (according to Theorem IV,) tf— 2^ 
fc=4, 2ZL. = ts =* 3, abz=A^ confequently iss=9, 

and fo iss + tf^=: S = i;J Wherefore | + jx~ 

+ 



Chap, via hy Infinite Seriei,' $9 

tHcn4^ g;=»xAxl5=B,tx^=fxBxii 
3= C, &c. AMb i/f X T^j = f, WHertfore IVf + 

L-^Jt ~ J^ 3- Thercfi>» L -^ +A+B4.C+D, 
Igc. ss L 3. See the Opera^on following. 

. L-^ = -^ ss . .. . : 0.4515449934959 

. X J = A is .....;. . 255467342296 

xxjs xAjt:B=s... .. 294656680 



Thu5i 



4xs-axC=t:D=...:; 15" 



s 



The Sum is the Ix^. of . . ... 3=o;477i2i2547*9-o 

' 11; But this Logtmthm may yet much fooner 
land with Ids Trouble be found, by the Artifice ufed 

in Art. 7. For the Ratio -2^ = fifil = J, and 

fo a4-^=s=65573, and <^--^=ss=37 » ^'^ ^"^ 

2^ X 4t = ^. therefore Li^i, 4. L 4, = 

5x3' z" 3* 5x3* ' 2 ' 

L3»i butL^ = Lz^' — L5. AndL^ 

f L2"— L5= ;;.;.. i 3.81647993061 
Therefore j 2«/ _. ; . , . . 0.00049010708 

The Sum is the Logarithm of 3» = 3.81697003770 

And * thereof is the Log. of3 = o.477i2i2547ii 
thus the J?f^ Step of Theorem 11. gives the Loga- 
rithm true to xi Places^, 



6o Ofcdfi^ruSifigLoGAitiTBMi 

12. The nextExample Hull bediat which Dr. 
Haffey has ^vea for finding die Ijogarithm of 23, 
which is done by Theorem IV. Indiis OUe^ 0=22, 
^24, ls=:23, i8S=:529y 41^528, and is^ak=: 

.0S7=S. Andf + jJ+^^. &c=iy|. 

»" "5 - 73 - P "!«'^ ^ ^' + 

L . '■ ', = L -r f andbecauie 2x2x2x3 = 

24, aml2xix=22, therefore aiib 3L,a4-L3 

= L245 andL2+Lii = L22. And^^^ld^ 

= L ^<ab I therefore, (proceeding in the Opera* 
tion according to Art 10.) we have 

■ — ^ ^" =s . . I.36i3i696u669o6i2945oo9i7a6698os 

"p ssAsf. 41087462810146814347315S86368 

7^ 7; ^ AsaBasa lasj 85 a 1 5441818^94^0074 

^ X i X B=C= 65832351 8437617$ 

^^S''^*^^' 4208829765 

' SS "■=^— »930 

fie Sum = L 23 =r . . i.36i7?7836oi7592g78867777, ,2,5, ,7 
which Logarithm is trac to 32 Places of Kgures, 
and thus you may proceed for aoy other. 

13. In making Logarithms for Prim Numbers^ 
the Artifice, otgreateft Mvantage tonfifts in finding 
ixxh^Ratte or Framon, whofc Terms are thtgreatefi 
poflible and their Difference the leaft j andthc Num- 
ber whofe Logarithm is fought, or fome Power there- 
of, IS an Ati^tot Part or Stdmehiple of one of the 
Terms of the &id .Rati* or FraaioH. Fot thh once 
obtaia'd, the Logarithm is (boa acquirad by The*- 

rem 



Chap.Vltt. ty Infinite Series, 6i 

rem II. with cafe. Thus the Fraaion |Ui « 
§iii2i »: i- } whetei=i, «&d 8=4641 the Series 

will convei^ very fwift for the Logsrithm of '*'*"_ 

L29— Li I added to theSerien, gives the Logarithm 

of 211. But the Raii^ or FraSion \Wltl. =ai 

53 97040 

tox54X29X43 ^'^ niakc the Scriesconvcrge very much 
iboner than before ; for here dsi, and 8=10774081. 
For L8o-hL54+L29-+-L43^Li2i added to the 
Series (or Theor. II.) gives the Logarithm of 211% 
half which is the Logarithm of2it. Laftly, the 

'wcaon— y^rfJJTo— 60x18x55x113x197 = 7j^*^^ 
i/^i, and J=396423888i, converges to thatDegree 
that the firft Step of the Series quotes the Logarithm 
of the Fradion to 29 Places^ to which add the Lo- 
^ithms of the 5 Numbers in the Denominatar^ and 
It ^ves the L(^arithm of 21 1^, then i of that is the 
Logarithm of 21I9 as before. 

14. The greateft Difficulty confifts in finding out 
proper Numbers for jAtxlucing fuch Fradions as 
aforefaid ; and the beft Method of this is by prudent 
fryals. An Example of which is here fubjoin*d. 
Suppofe I would procin^e a convenient Fradion for 
the Logarithm of 223, I ioiake tryal thus i ' 

Therefore 223 X387 = 64001 

Having thus obtained the Term 64001, 'tis cafy to 
ob&rve the other may be 64000, wherefore the Frac-^ 

tioo is imi » !!Z^^ and finds the Log^ithta of 

223 



6i Of conjlhi&ihg LogaiIithms 

(223 with good Difpatch. Or thus, to find a Frac- 
tion of larger Terms \ fuppofe I afliime the Nume- 
rator 159000=1000x53x3, then to find the othct 
Term as near this as may be, I try thus ; 

223^ 3= ^^9 

^CX 1 = 223 

^^3lx 7 =1 561 
Therefore 223 x 713 =ii 58999, which, is withiA 
tJnity as great as the other Term 159000, and con- 
fequently the Fraftion WHH is that required, and 
thus you proceed to raife the Terms of any other. 

.15. Let the Terms of ah)r Fr^dfidii be reprefented 
by ^^^Leaft, and ^=Grcateft, Then if the Ratio ht 

intreajing itv&llbo ^, but if decreafing, i.; let that 

Term,- in which the Number fought is ingredient ^ be 
cxprefs'd by the Produft rx, where r= the Number 
(or Ptoduft of Numbers) whofe Logarithm is kriown^ 
and j^=: the Number whofe Logarithm is fbtight. 

K a=cxj then ^ = 4> <>r "~ > \sat\ib=cxy thcli 



ex 

— • or — 



J, or ^ ; alfo let there be put the, ad Theorem 
?l+i^ + g + ii^ fSc. .= z ; then if the un- 
known Number x be in the Denominatbr of the /»- 
creafing Ratio jy viz. ^ j or in the Numerator of 

the decreaftng Ratio ~, viz. — ; then it will be 

Z+Ltf — L^=LAr. Andy vice verfa, ifitbe ^ or 

A, the Theorerii will be Z+lJ-^hc=l^. Ftom 

ifierice the Operations in the foregoing Articles for 
making Logarithms haye their. Grounds and Reafon ; 

4ftd 



Chap. VIII. by Infinite Berks.' 63 

and every thing there aflerted is from this Procefs 
very evident. 

16. To find the natural Number of any Lc^- 
rithm propofed •, this is beft done by the Thcoreta 
in Art 17. Chap. VII. 

For Example, let it be required to find the Intereft o£ 
one Pound for one Day, at the rate of 6 /. per Cent, 
per Jffnum, Coatpound Intereft % which is to extraft 
the Root of 1.06 taken as the 365th Power; thus 
Sic Logarithm of 1.06=0.025305865^2647702408- 
46731 1 8635 1, ^c. 
Whidi divide by 365, 1 £=0.00006933 1 1377 1 16- 

the Quotient is J 592899910443^^6, (^c. 
The next neareftl fc=|.oooi6=o.oooo6948i5587- 
' Log.&itsN^l 28037517724712696, £«?<•. 
Their Difference is 3=0.000000150421016338227- 
733668350, fcff. 

Mult, this by »=2.3025850929940456840i799i4- 
54684, &c. 

TheProduftis 
»a =?= 0.000000346357189893416971322305 . 

119963302990,864503 

4i550»525H 

14391 

The Powers of n a. Then 
i-f4«*a*=*i.ooooooooooooo5998i65i49543225i 

/4»*a*;= 599 

The Sum iH-i7»»S»4- \ t .00.000000000005998 1 6$- 
A«.*a*=' 3 i49543285i==X. . 

The odd Powers 

{»a= 0.000006346357189893416971322305 
i»5a"'= 6925025419 

Sum »a-H»'3*=o.oooooo346357i 8989342389634. 
7724=z. 

Then 



f »a =?= 



64 7& Ijgarkbmk Sfdral. 

Then the Value of the Series ia 

X-«=o.99999965364287oo88227599o85 1 2^. 

Which multiply bf ^==1.00016, produces 



t»f 



jnthsn L> and that to 30 Places of Figures. The 
fime Number may be feen produced to 60 Places in 
Mr. ^Amcra^Mathea^tical Tables. 



CHAP. DC. 

{^ the ^IjKtaritbmc ^Spiral I and the Nafun 
0nd C§nfiruBion of the TaAk rf "^Mtridimal 
*Parts^ or tbe Nautical ^Meridian *Line^ 
deduced therefrom. 

i.T F any Right-Lim^ ^W be moved with an e- 
X fioble Motion at)out the fix'd Point />, and at 
the iame time the PcMnt W be movM towards the 
Point /; with a Velocity fuch that the Radii pVf, pV^ 
fS» (^e. form'd thereby, be in a Geometrical Ratio 
decreafing» then the Curve W VSQ» (^c. is called 
the Logarithmic Spiral ; and diat for the £une Rea- 
Ion as the Logaridimic Onrve before defcribM received 
its Appellation. See Chap. III. Art 8, 9, la 

a. For fuppofe the Arches A(S=£E=EG, dfc. 
and therefore m Arithmetical Progreffion $ andfince^ 
from the Generation of the Spiral^ the Radius 
pB:pD r.pD:pF::pF:pH»^c.*tisevidentthe Arches 
AC» AE, AGt tSc. are the Exponents of the Ration's 
oi^Raiii Dp» Fjpt, Hjp, 6fr. to thefirft pB ; and 
fo thpf? Arches are m refpeft of the Radii ^ as Loga^ 
ritbwts VBLVtfyeSt of Numbers ; as is fufficiently mani* 
feft from the preceding J}>eory of Logarithms. Where- 
fore 



Chapik. 7le LbgaHthfiic Spiral. 65 

fbreifBpbc=i, 10, ioo,f5f^.aiidPW(thc lOthPro- 
portiooalfroxn pB) be=io, IQO, 1000, &c. then (hall 
the Arch AC=, 1000000, AE =,200000, AG =, 
3000000, £sfr. AW=i ,0000000 ; be the Loga^ 
rithras of tte Numbers pD=i, 259, (^c, pF=i,585^ 
C^'r.pHzi:! ,996, fcf c. pWzrio j of Mr. Brigg^s Form. 

3. This Spiral is alfo called the Equiangular Spi-^^ 
ral i becaufe it interieds all the Radif pW, pQ,pB» 
at equal Angles. For fuppofe the Arches NP, TWi 
infinitely fmall, and equal to each other, then may 
the Parts of the Spiral DQ^and VW, be cfteemcd 
Right-Lines; and i(b fince m the Triangles pOQ. 
pV W, the Sdes are proportional, v/e. Op : pQ^:; 
Vp : pW, and the Angle OpQ^VpW, thofe Tri- 
angleis. are fimilar ; and coniequenily the Angle 
pOQ^pV W^ br pQp=pWV 5 and thus it will be 
every where. 

4. Now let the lyhole Scheme Be cdnfidered a9 
the Sterasgrapbic Projedicn of one Qjurtcr of a pe^ 
tallel Hemifpberey then fliall p be tbeP#/^ 1 WL A, 
g Quadrantal Arch of the Equktor \ the RaMi p W, 
pT, pR, iSc. the feveral Meridians projcfted on the 
Plane of the Equator. And fince 'tis the Property of 
tvery Rumb Line to make equal Angles with everf 
Meridian oh the Globe, and the Angles contained 
between circular Arches on the Globe, are equal to 
the Angles between thjB fame Arches in this Projec- 
tion, therefore the Logarithmic Spiral WQg is the 
Projedion of a Rumb Line \ fince it has the fame 
Property on the Projeftion, as the Rumb on the 
Globe, as was proved Art. 3. hereof. 

5. Moreover, fince all Right Circles^ fuch as arc 
rtie A^ridiafts in this Cafe, are projeiSed into Rigbs 
Lines equal to the Tangents of half the Arches, the 
Lihes pB, pD, pF, pH, i^c. will here reprcfent the 
Tangents of half the Complements of the Latitudes 
AB, CD, EF, GH, ^c. And fince the Arches in 
the Equator AC AR, AG, ^c. are the Differences 

^ K • «/ 



66 7}>e Logarithmic Spiral 

of Longitude made by failing from the Latitude B to 
the Latitudes D, F, H, ijc\ on the RunA or Spiral 
BOW; and it has been fhewn that thofe Arches 
are the Logarithms^ o( the Radii ^D^ pF, pH, ^c. 
therefore the Difference of Longitude is the i-^tf-. 
Hthmof the Tangent of half the Complement of Lati-^ 
tudcy reckoning from the Meridian A p whence the 
Logarithms begin. 

6. Therefore the Difference of Longitude RT, 
made by (ailing from the Latitude $ to the Latitude 
y, is equal to the Difference of the Logarithms 
(AT— AR) of the Tangents of the half Comple- 
ments (Sp, Vp) of the Latitudes TV, RS. Andl 
fince the Ratio of the Progrejfton^ or of pW to pV,* 
may be infinitely varied, 'tis plain the infinite Num- 
\xx of Rumbs in a Quadrant of the Compafs deter- 
mine fo many Scales^ Logarithms in the Equator of 
^he Tangents of the half Complements of the Lati- 
tudes proper to thofe Rumbs. 

7. Since then every different. Rumb is a Logarithm 
inic Spiral J or determines a peculiar Scale of Log0^ 
rithms for the- Tangents of the Half-Complements of 
its Latitudes,' therefore any Canon or Table of Lo- 
garithm-Tangents, whether of Jfeper^s^ Briggs\ or 
any other Form whatfoever, is the Scale of the Pif- 
ferences of Longitude on fome determinate Rumb or 
other. And confequently if this Rumb be invefti- 
gated for the Canon of Brigg^s Logarithms (now la 
common Ufe,) the faid Canon may be made to'an- 
fwerall the Purpofes ot tht Nautical Meridian Line^ 
in Propofitions of Navigation by Mertatofs Chart. ^ 

8. In order to this it muff: be confidered, that the 
Meridian Line is a Table or Scale of Longitudes to 
every Degree of Latitude on the Rumb which makes 
an Angle of 45 Degrees with the Meridian ; fince 
in this Cafe the Differences of Longitude are always 
equal to the Meridional^ or enlar^d Bifferemes of 
Latitude. And fince there % a certain Rumb oa 

which. 



chap. IX. ^^e Logarithmic Spiral. 6j 

which Neper^s or Brigg^s Logarithm-Tangents are 
the Differences of Longitude, and the Differences of 
Longitude on different Rumbs are to one another as 
the Tangents of the Angles of thofe Rumbs with the 
Meridian ; therefore by having given theDiiFcrcnce 
of Longitude on theRumb of 45°, in Logarithms 
of Nepet^s Form, and the Length of the Arch of one 
Minute or Degree in Parts of the Radius, we can 
dience find the Angle of that Rumb which deter* 
mines that Species cf Logarithms. 

9. Now the Momentary Augment or Fluxion of the 
Tangent- Line of 45®, is exaftly doubJe to the Flux^ 
ion of the Arch oif the Circle (as is eafily proved), 
and the Tangent of 45'' being equal to Radius^ the 
Fluxion alfo of tht Logarithm-Tangent will be double 
to that of the Arch, if the Logarithm be ofNeper^s 
Form i but for Bri^s*s Form, it will be as the fame 
double Arch multiplied into ^0.43429, &c. or di« 
videdby »=:2. 30258, &c. the Index for Briggs*sLor 
garitbms. See Chap. VL Art. 7. 

10. l^Tow fince the Radius of a Circle being put 
=*=i, the Periphery thereof will be 6.2831853, Gfr. 
therefore 360)6.2831853, £9^^.(0.01745329, (^c. =s 
the Length of the Arch of one Degree. Alfo 
60)0.01745329, &c. (0.0002908882, &c. = the 
Length of an Arch of one Minute^ in Parts of the 
Radius. If one Minute be fuppofed Unity, then the 
Proportion for finding the Angle of the Rumb 
required for Neper^s Logarithms, will be, as 
I : 2.908882, &c. :: Radius = loooooo, &c. : the 
Tangent =2^08882, &c. of the Angle 71' 1' 4^'^ 
whole Logarithm is 1Q.463726117, &c. and under 
that Angle is the Meridian interfe(^ed by that Rumh 
Line, on which the Dirfercnces of Neper*s Loga- 
rithm-Tangents of the Complements of the Latitudes 
are the true Differences of Longitude, eftimated in 
Minutes and Parts, taldng the &rft 4 Figures for 
Integers. 

K 2 lit 



68 CofifirueimoftbeT^aik 

%i. But fince Neper^s Logapthms are to thoA: of 
Mr. Briggs\ Form, as 2.30.2585, &c. is to i .ock^ooQij 
&c. therefore to find the Angle of the Rumb ifor the 
logarithms of Briggsfs Form ; this muft be the A- 
nalogy. As 2302585, &c. : 2908882, &c. :: loooooq 
= Radius : 12633 114, &c* =^ ^^® Tangent of the 
^"gl^fi" 38'9', whofeLogarithpx 1810.101510428, 
&c. Wherefore in the Rumb Line that makes an 
Angle of 51** 38' 9'' with the Meridian^ the commotit 
(viz. -Br/gf j's)X#ogarithm-Tapgeiit§ are the true Pif- 
fcrences of Longitude. 

, 1 2. But if a Table or Scale ofJJgarilbm-'itangenis be 
made by Extradion of the Root of the infiniteft Pirw- 
eVj whofe Index i$ the Length of the Arch you put; 
for Unity in the faid Scak •, then fuch a Scale oT I^- 
garithm-Tangents fhall be the true Meridian line re- 
quired. If then the Radius or Tangent of 45*, be 
put = I ; and the Diflirence between Radius and 
any other Tangent T, be called t; fo that it* be 
R±t=T ; the Logarithm of the Ratio of Radius to 
fuch Tangent, will be 

Logarithm of the Tangent T, when it is R+t=T. 
OrJxt+CH^ti +!! 4.1:1, (^c. when if: 

2 3*^4^^5 

IS R — t=T. All which is evident from Chap. VL 
Art. 6. 

13. According to the fame Dodlrine (Art. 9. of 
the fame Chap.) if T be any given Tangent, and t 
the Difference thereof fi^n^notj^LTangent ; thei» 

the Logarithm of their Ratiowlnsf^ ^ r^-f* 

\^ -ir- + jh^ ^'- ^^^'^ ^ « thel&eT 
Term. But 

when T is the greater Term, 14. 



jphap. IX. ^Meridional Parts. €^ 

14. Again, it was (hewn in the ikmc CSiap. VL 
Art. 10, and 1 1 . that thisScrics may be made to con- 
verge twice as fwift, omitting all the even Powers^ 
by putting 5^= the 5'«w of the Tangents, and t as 
the Difference J as above. For thus the Logarithni 

garithm of the Ratio of thofe two Tangents. 

15. But the Ratio of T to t, or of the Sum of tw0 
Tangents to their Difference is the feme as that of the 
Sine rftheSum of thofe Arches to the Sine of their 
ipifFerence ; that is, again, as the Rado of the Ctf- 
Sine of middle Latitude (or half Sum of the Arches) 
to the Sine of half the Difference. Therefore putting 
S = Sine-Complement oi middle Latitude •, and s for 
the Sine of half the Difference of Latitudes > thea 

i- = — i and fo the Series will become -; % L jl. 
iL. X iL 4- iL , fc?r. wherein as the Differences 

pf Latitude are fmaller, fewer Steps will fuffice. 

1 6. So that, if the B/puitor be put for Middle-La^ 
titudcy then Ihall SrsRadius, and /=5ine of the La- 
titude ; then the Meridional Parts reckoned from the 

Equator will be ^ x f + il + 41 +^,,6?r. 
Here bccaufe r=i, therefore ^x^-f-ll^-ii^ 

C, fffr. the half of which is i + '1 +.C+ C.^ 

0?r. = half the Logarithm of the Ratio of r +• j to 
r — S'y that-i^yrfitb^ verfed Sines of the Diftances 
from both Pole$. See Chap. VI. Art. 1 1. 
i -17. I (hall exemplify this Series by ihewinghow 
the Meridional Parts anfwering to 30* Latitude, wre 
to be found thereby, and that by the LogU'ithms^ 
as follows. 

The 



Multiply by ..•.,.-.,.., *. 3 

The Logarithm of , . . . j':?: .9.0969106 

Subftrad the Logarithm of 3= 047712 13 

There remain* thpi s^ ^^.,ic/:^^ q ^,/^l*Qo•. 
Logarithm of I 7 = 0.0416657=8.6197887 

Andprocecdingthus, [7 =o.oo62500=.7.79588oQ 
you'll find the o- x7 ^ ^ 

ther Steps of the] ^=0.0011160=7.0476920 

i. =0.0002 1 7I=:.6. 3364875 



Series by their Lo- 
garithms, as here 
fct down. 



^'— =o..oooo444=-5-6472773 



TheSumofs-+- fl + T 

f + r + gT + [=o-5492942=.97398o5V 

ii:, &c. is \ 
11 J 

To which add) * ^ « , 

the Log. of|i^34377467707g>&c. =3.5362739 

TThe Sum is the-j 

Log.oftheJk&- I 

ridional Farts \ itfa:. 1888,334, &rc. =3.2760790 

for the Arch I 

of 30^ J 

18. And thus you may proceed to find the Length 
of any other Arch, or the Diftance from the Meri- 
dian of its Parallel of Latitude ; ^nd fo the Meri- 
dian Line may be conftruded de no^ifo^ if any one 
thinks \t worth while. But tho* it may be dpne with 
greater Accuracy and Exaftnefs by thcfe infinite Se^ 
ries than what wc have by the common Method ; yet 
the Table of Meridional Fart 5^ or Nautical Lin^ ', 
made from thence, now in Ufe, is abundantly fuf- 
iicient for all the Purpofes of Sailing *, and confe- 

quently 



Chap. IX. cfMeridmalParts^ "^ f% 

Guently renders a new Calculation thereof unaecef- 
llaiy, and a Matter of mere Curiofity. And indeed^ 
fincc it has been (hewn above (Art. 9, 10, 11.) the 
Meridian Line is no other than a Scale of the Loga* 
rithm-Tangents of the Half-Complements of the 
Latitude on the Rumb of 51*38' 9', the Propofi- 
tions of Sailing by this Method are refolvable by 
only the Canon or Log^rithm-.Tangcnts at the End. 
of this Treatife ; fo that where this Canon is at hand, 
neither Meridional Table or Line can be neceflary, 
as will appear by a Chapter particularly on this SuIh 
jeft, in the Prallical Part. They who would fee 
the Thepry of this Branch of the Art^ may perufe 
N**. 219. of the Pbilofopbical Tranfaffionsl where 
they will find a moft learned Traft on this Subjeft, 
wrote by Dr. HaMey 5 from whence the Subftance of 
this Chaptier is taken. 



C H A P. X.' 

Oftbe Conftrufltlon of a Large ♦Logarithmic 
Scale, €xbibiting by Infpe<aion a diftincS * I- 
dea of the Nature and Agreement of^ Num-^ 
bers and their * Logarithms. 

1. T T is an Obfervation of the earlieft Antiquity^ 

J^ that wc have no Ideas in the Mind whicK ^^ M 
were not firft in the Senfes -, or that the Senfes of the, ^ ^ 

Body are the only Inlets or Entries by which the Idea*s^ ^. ^ 

qiObje£ls prefent themfelves to the Mind. It fol- • 

lows then, that the Idea*s muft needs be fo much the 
more clear and diftinSim the Mind^ and confcquently 
be the better underftood by it, by how much the 
mote fully y compleatly^ viud oivioujly they firft of all 
affe(St our Senfes. Single uniform Objc(5l8 cafily in-, 

^ linuate 



ji TBe Conftrudlton ofd Scale: 

llnuate themfelves, and make ftrong and clear Ini- 
preflions on the Mind^ while thofe which involve 
Multiplicity and Variety in their Nattire, are pro* 
portionally more difficultly apprehended by thd 
Senfes, and confequently afKsd the Mind with im* 
ftrfeEly confufed^ znAJUghi ImpreJJims^ which there- 
fore muft produce a moreperplex^d^ eh/cure^ and un^ 
eertain Noticn oi Conception of the Things them^^ 
felves. 

2, From this G>nfideration we may eafily learn 
tiieReafon^hy, of all the vaft number ofPerfonsi 
who underftand the praSical Ufe of Logarithms, fa 
very few of them know any thing of the Nature and 
ConftruSfiott of them.. The Ufe of Logarithms is 
rerv obviogs to the Senfes by eafy Examples, but 
their Nature and Conftruftion lead the Mind too 
much upon the Contemplation of Infinities both o^ 
^antity and Variety j which are Subjeifts too vaftly 
^firufe znd remote from Senfej ever to be verv com- 
monly underftood 5 unlefe fome Expedients oe con* 
trived, which may help to facilitate fo difficult ail 
Aflfair. 

3, And as there are principally but three Ways^ 
whereby the Nature ot Logarithms are explained^ 
^iz. by Numbers^ Species^ and Lines^ the Expedient 
albre£ud muft be fought in one of thefe three Me- 
ibods. But Numbers^ of all things elfc, exhibit thfll 
mod: complex and various Idea^ therefore it cannot 
be hoped for from them. Species^ on the other hand^ 
«re too Jimple and concife a Reprefentation of fo vafi 
and various Ideals as are thofe of Logarithms^ and 
have nothing of their Refemblance in their Form i 
tills Expedient therefore is not to be expefted from 
diii Head. It remains then, that Right or Curvi 
tints be ufed for the Purpofe of explaining the Na- 
ture of Logarithms, by making the whole Matter ob^ 
viousto the Senfes. And here indeed we ihall ^xyi 
ail that can be deiired, or is necefiary to the Purpofe. 



Chap. X. ^e Con/iruaion of a Scale. 73 

4. For Example, let the complex Idea of a tbou- 
fond Units be to be exprefs'd moft advantamoufly to 
a Mind unexercifed about fuch compleifd Notions ; 
if you do it by Numbers or Figures, it muft be by 
this Expreilion 1000 ; but there are but four Cba^ 
raSers to form an Idea of a tboufand Separate Objeds 
in the Mind. In Species, this great and complefc U 
dea is often reprefented by one Charader alone, as ^ ; 
or two, as i-f-x ; which are ftill more obfcure and 
abfblutely unintelligible without fome PrerNotions of 
the Matter. But a Line may be taken of a Length • 
fufficient, that by proper Divifions, all the thoufand 
Units may be rendered diftinft and <^vious to the 
Senfe, in any variety of Magnitude almoft, butefpe* 
cially if they are equal to each other, as in the Cafe 
of Logarithms before us. Wherefore, fince by Ldnes 
fuch great and almoft inconceivably complex Ideas are 
capable of being reprefented to the Senfes, diftina 
and feparate in their proper Parts j and the DoSlrine 
of Logarithms depending entirely on fuch IdeaSy 'tis 
evident that by Means of Lines of afuf&cientLength, 
the Nature and Properties of Logarithms, and the 
Operations thereby, may be rendered more apparent 
and compleat to the Senfes, and fo be better Mf^irr- 
fiood in the Mindy than by any other Means what* 
foever. 

5. The Confequence of all which, is, that the 
young T^yrOy and all who would have a true Notion 
and moft clear Underftanding of this abftrufe and my^ 
fierious Doftrine, (hould be affifted with fuch a large 
lineal ConfiruSion of the Logarithms^ as hath been 

. hinted at. And this, I hope, I have efFcdlcd in the 
large Diagram on the Copper- Plate before you, with 
confidcrable Exadnefs, which I call the Logarithm 
mic Scale. Wherein all that has been faid in the Ge- 
neral Theory aforegoing, or may follow in thtpr api- 
cal Operations of Logarithms^ is evident even to 
Senfe itfelf^ to a very wonderful degree, by a bare 

L4 Itilpedlion, 



74 ^^ QmftruSHon of a ^cak. 

Infpeftion, or a Glance of the £y^ only ; and th«re<« 
fore cannot but conduce to form a very diftinA and 
agreeable Idea or Notibn both of the Theory and 
Praxis of this admirable Art. 

6. The Scale confifts of three principal Lf^eji 
which bound it ; thtjirjl is AB on the Side^ which 
is 22 Inches in Length, and is divided into looo e-^ 
qual Parts which repreiient the naiurd Numbers from 
1 to I POO, all which are viiible and diftinA to the 
naked Eye % which Numbers therefore are affixed to 
every ibth Diviiio^. The fecond is AC at the 
3ottom, divided into 300 equal B^ts^ (as being but 
I Bts Inches long.) Thefe reprefent the L^ariwms % 
if each of thtftequ^I Parts be fuppofed to reprefent 
10, or 100, the Logiirithmsy then^ for all Numbers 
under 1000, will be exhibited by Lines only to 4 
or 5 Places of Figures, including the Indicts. The 
li&/y^ principal Line is the Logaritbm^tic Curve 
cDEFB, in which all the Lines of iV«w*frjand X^- 
^arithms terminate, and whofe Genefis and Proper.^ 
ties have been before 4cfcribed. See Chap. Ill, IV, 
V, 6fr. 

7. The Scale confifts (or is made up) of Lines of 
Numbers, and others which are the Complements of 
the Logarithms to 3.0QOO, The firft ^reperpendi^ 
cular to the Logarithmetic Line AC in its fcvcral 
Divifions, and increafe in Length in a Geometrical 
Ratio ; as hath been obfervcd : thus dividing AG 
into 3 equal Parts CG=pGH,=HA, if GD be the 
loth proportional Term from Cec5=i, or Unity, then 
ftall GD=ioCe=io, and HE=ioGD=iooCe?s« 
100. Laftly, ABf^ioHEs=iooGD=ioooCc=: 
J ODD ; as is evident from the Nature of the Curve^ 
and by InfpeSiion. The Complements of the Loga-^ 
rithms are the Lines which run acrofs the Diagram^ 
parallel to the Line of Logarithms AC ; thcfe at the 
Curve refer the Numbers to their proper Logarithms, 
and by means of thofe Lin^a thus croffing each other 

in 



Chap.X* TieCcnJiruSionofaScak. - 75 

iA every Part of the Scheme^ the Logarithms of Num-^ 
her^ and the NtLWibers oi l»aritbms art moft eafily 
&hd obvioufly foundi for mt Extent of the Scale^ 
by Infpeaion only. 

8. It is not pretended that this (or any other) In^ 
ftrument is capable of ahy great £xa)ftnefi in prciHtal 
Operations \ 'tis fucflxient formy Deiign, ifitoiily 
illuftrates void proves the Truth of every part of the 
DoSifine of Logarithms to the Senfe^ and thereby 
tenders it eaiier to the InteUeSs of young Learners. 
If the Tbitnry before delivered be examined by this 
Scale ^ it will be found to agree with it to zfenfible 
ExaSnefs \ it being as it were but the fame thing at 
large. In the following Parti I ihali (hew the Cor- 
tefpdndente and matud Agreement bdtween the fun- 
damental Operations by Logarithms wrote by Num- 
bers^ and the fiune performed on this Scale ; than 
Which nothing more^ that I know of« €an be faid of 
expeAed., 

G H A p. Xt 

Of the Conftru6tion f>f the Artificial hi^t^ bf 

g/^/jJ^ Logarithms; 

< 

I. np H E Canon of togarittms being cotopleated 
X «WJd orderly digefted in Books ^ tho* this was 
a greater Adv.antage than the Mathematicians of any 
former Age enjoy'd, yet not content to have a bulky 
Book of Logarithms, fit tp be ufcd, in Studies and 
with the Pen only^ the reftleis and unfatisfy'd Fa- 
culty of Invention in Men put them upon Contri- 
vances to nc^-model and reduce tljf voluminous Art 

La to 



^6 Cot^ruBion of Artificial Lines 

to Mniature^ that fo it might be renderM more 
eafily manageabk, and more univerfaUy ufeful. 

2« la the Purfuit of this De(ign they alfo very 
well fucceeded ; for (ihce Numbers of any kind are 
capable of being reprefented by Right- Linesj they 
were not long unappriz'd that the whole Body of the 
Canon oi Logarithms might belaid down and ex- 
prefs'd in tbie Diyiiions of one ftralt Line. Mr. 
Gunter^ Profeflbr of Geometry at Gre/bam-College^ 
was the iirft who took this matter in hauid, and con-^ 
ftru^ted fuch an ^/f^/WL/»^ of Logarithms ; which 
therefore from him was called (ever fince) Gunter's 
Line^ orfimply, t\xtGunter. The (amePerfon alfo 
conftrudted artificial Lines of Sines and ^Tangents ; 
and ail thoie Lines, with fome others laid down on 
a Scale, make what we commonly call Gunlet^^ 
Scale. 

3. The ConJiruSion of thofe artificial Lines is 
eafy to be underftood, and is as follows. Draw the 
Rtght'Line AB (Fig. VII.) which divide into 10 
great equal Parts, as is there denoted by i, 2, 3, 4, 
&r. and •each ofthefe into 10 others, and foon. 
Conceive thefe feveral Divifions^ or equal Parts^ to 
reprcfent the Logarithms in the Canon for the natu- 
ral Numbers. Now fuppofe the whole Length 
AB=io, then the firft grand Diviiions will be i, 2, 
3, 4, &ff. But if ABr=ioo, then the firft Divijions 
will be 10, 20, 30, Csff. and the fecond Divijions 
i> 2, 3, (^c. Agairt if AB=iooo, tht prime Di- 
vijions will be 100, 200, 300, &?r. and thtjicondary 
Divifions, 10, 20, 30, 6fr. Suppofe the latter Cafe, 
viz. AB=iooo; then draw another Right- Line CD 
e^al and parallel to AB, the natural Line of Lo- 
garithms. 

4. Now in the Line CD, fuch Divifions are to be 
made as may reprefent the Places of the natural 
Numbers i, 2, 3, &?r. or 10, 20, 30, 6?^*. or 100, 
200, 300, &?f . But neglefting thtindices of Loga- 
rithms, 



Chap. XI. rfNumbtri^ ISines^ ficc. 77 

f ithms, *us plain, fince the Logarithms of the Num- 
bers i» 10, 100 % a, 20, 200 i 3, 30, 300 1 &c. 
are the &me, the Dtftances of thofe Numbers will be 
the fiime on the Line or Scale CD. And there&re 
fince the Lc^rithm ofi, issaso, the Number i 
muft be placed at the very Beginning of the Line 
CD, from wheiKe the Logarithms begin in the Line 
AB. Thenbecaufe the whole Line AB <ss 1000 tzr 
Logarithm of io=CDy therefore againft the Loga- 
rithm of 2, which is ^^sgoi in the Line AB, mSco 
a Divifion in the Line CD, and by it place the Num- 
ber 2. Again, becaufe the Logarithm of 3 is 2=47 7 
in AB, therefore torrefpondent to d)e Point in AE^ 
m^c another DiviGon in CD, and by it place the 
Number 3. The Logarithm of 4 is 602, therefore 
from 602 in AB make a Divifion in CD, by which 
you muft place the Number 4 ; and thus you pro- 
ceed to find the Divifions for the other Numbm to 
lO in the Line CD, by the Logarithms of thoie 
Numbers in the Line AB. 

5. If the Divifions in the Line CD now found 
for the Numbers i, 2, 3, 4, &r. be fupposM, in«> 
ftead of them^ to be for the Numbers 10, 20, 30^ 
40, &r. then each of thofe Divifions may be fub- 
divided into 10 others, by the Logaritbmic Parts 10 
the Line AB. Thus, because the Logarithms of 
II, 12, 13, I4i 6?r. are 41, 79, 113, 146, iSe. 
therefore againft the(e latter Numbers in the Line 
AB, make Divifions in the Line CD, fo ftiall the 
firfi: grand Divifion from i to 2 be divided again into 
10 others. Agam, becaufe the Logarithms of 21, 
22, 23, 24, ^c. are 322, 342, 361, 380, i^c. 
therefore Divifions made in CD againft thefe Num«* 
hers in AB will finifti the Subdivlfions of the Space 
from 2 to 3, in the feid Line CD. And thus pre- 
ceeding, you may fubdi vide all ih^ other grand or 
prime DivifionSf to the £nd of the Line. 



7S Cm^Bim i/ArtiJkialLinei 

6\ If your Lines be of fd great Lengthy tluC Oititt 
kft Sabdivifions in CD, aTfe ftill of aUngth capabte 
of another tmfdd Divifion \ then the firft grand Di^ 
vifioos muft be reputed loo, 206, 366, &r. and fo 
fince the Logarithms of the Numbers loi, 102, (Se. 
201, 202, ^^. 301, 302, 6fr. arc 4^ 8, Cs?r. 303, 
305, &r. 478, 480, &r. therefore ' if againft thefe 
Parts in AB, you make Divifions in CD» there will 
enfue a triple Divifion of the faid Line CD, which 
. is more than is neceflary for Inftrumental Ufes^ ahd 
indeed cannot be done but only for the two or three 
firft Diviiions. 

7. Thus have you (een the Conftnidtion of the 
Artificial^ Logariihnic^ or Guntet^s Line^ fo famous 
in all Parts of the Mathematics. A Line which per- 
forms the Buiineis of the whole Logarithmic Canon \ 
iince the Divifions of this Line l»ve all the fame 
Properties with regard to the natural Numbers on it, 
as the Logarithms of the Table have to the Numbers 
corrcfponding to them. *Tis plam the Divifions and 
Relation of thefe two Lanes AB, CD, are the fame 
as At, and AT, in Fig. IV. and III. Confequently 
what has been ilaid of thofe Lines heretofore will help 
to lUuftrate the ^eory and Conftruftion of the Lines 
now under Confideration. But fince in Ufe the 
Gunter CD isfuppofed to be divided into an lob 
Parts at leaft, therefore you always (or moftly) ob* 
fcrve it 6f a double Length of that which is exprcfled 
in Fig. VII. which Length is commonly called 22^- 
dius % and fo the Gunter in common Ufe is iaid to 
be of a double Radius ; becaufe elfe the Divifions for 
the nine Digits would be wanting, fince the Diftance 
from I to 10 is eaual to that from 10 to 100, as is 
evident from the foregoing Conftrudion, and from 
the T'A^ory of Logarithms. 

8. Having thus fliewn the Conftrtt3ion of the Line 
cf Numbers y the Con/iruaicm of the Lines of artificial 
iines and Tangents cafily follows j fince, as before 

.. obfervcd, 



Chap. XI. ofNumbiTSy Sines^ ficc. 79 

pUerved^.the Logarithms of Sines ztidTangents are 
nothing more thuk common Logariibmsoitacli Num« 
|l)ers as exprefs the Sines and Tangents of each Minnie 
gfitic^dranL 



Ocg. 


N.Sine. 


1-g. 


N.Tang. 


L<^. 


I 


17 


2418 


17 


2419 


, 2 


34 


5428 


35 


5430 


3 


52 


7188 


52 


7194 


4 


69 


8345 


1° 


8446 


5 


»7 


9402 


»7 


9419 


6 


|04 


1.0192 


105 


I.02l6 


7 


121 


1.0858 


122 


I.089X 


8 


>39 


».i435 


140 


I.X478 


9 


156 


i.«943 


»58 


1. 1997 


10 


^73 


1.2396 


176 


1.2463 


20 


342 


1.534® 


363 


1.5610 


30 


500 


1.6989 


577 


1. 76 14 


40 


642 


1.8080 


839 


1.9238 


50 


766 


1.8842 


1191 


2.0761 


60 


866 


1-9375 


1732 


2.2385 


70 


939 


1.9729 


2747 


2^389 


80 
90 


.984. 
1000 


1.9933 
2.0000 


Innn. 


^'753^ 
Infin. 



In the Hftle Table above, the firft Q)lumn contains 
the Degrees J the 2d and 4th the Natural Sines and 
Tangents^ and the 3d and 5th Columns contain the 
Logarithms of tjiofe natural 5i»w and Tangents^ the 
Indices being omitted^ and the Radius fuppofed == 
10000. 

9. Let three Lines be drawn, and let L ^ Line of 
Logarithms^ or double Radius of 20000 equal Parts ; 
S = a Line for Sines j and T = Line for Tangents j 
the two latter muft be drawn equal and parallel to the 
firft i as in theConftruftion of the Line of Numbers. 
Then having graduated the Line L mto 20000 equal 

Parts, 



8o ConJlfuBim of Artifitial Lines 

Parts, ifagainft Aich (^thofe Parts as are exprefsM 
by the Numbers in the 3d Column, you make Di* 
vi(|9ns in the Line S, and by thofe Divifions you 
place the Numbers in the firft Column, you will thefft 
have the artificial Line of Sines S graduated for the 
firft great Diviiions of i, 2, 3, 4, 5, 6, (^c. 10, 20, 
30, Gfr. Degrees : After the fame manner by the 
Table of Logarithmic Sines you find Numbers, fi-om 
whence in the Line L you findDinfions in the Line 
S for Minutes^ and Parts of Minutes. And thus the 
Line of artificial or Logarithmic Sines is finiibed. 

10. Again, from the fame Parts of the graduated 
Line L, as are found in the fifth Column oif the Ta- 
blety you make Divifions in the Line T, and by them 
place the Numbers of the firft Column, the Line T 
ihall be the artificial Line of tangents graduated for 
the firft great Divifions of i, 2, 3, 4, 5, 6?f. 10, 20, 
30, £5?r. Degrees. And the Subdivifions for Minutes 
will be found as before direded. But tho* the doUile 
Radius on the Line L fufiiccs for graduating the Line 
of Sines S, to th6 whole Length of 90 Degrees, be- 
caufe all Sines arc lefs than the Radius of a Circle, 
which is the greateft Sine ; yet becaufe the Radius of 
a Circle and the tangent of 45 Degrees are equal ; 
therefore 'tis evident the Logarithms of all Tangents 
greater than 45 Degrees, will exceed the Length of 
the Line L, as is plain from the fifth Column of the 
foregoing Tablet. 

1 1. But fince Radius iszmean ProportionalhtVNt^ti 
the Tangent of any Arcb^ and the Tangent of that 
j^frriV Complement, itfolfows, that the ;?^/«r4/7W»- 
gents in the Gecmetric Ratio or Scale zrt equally diftant 
on ^ach Side from the Radius or Tangent of 45 De- 
grees : and therefore the Logarithms of thofe natural 
Tangents^ which are equidijlant en each Side the Ra- 
dius or Logarithm of 45^, are alfo equidifferent ; that 
is, their Differences are equal. Jhus the Difference 
of 44^" and 46^, from Radius is the fame j and 

the 



Chap. XI. of Numbers^ Sines^ &c. 8r 

the DiiFcrenccs of 40^ and 50®, 30^ and 60^, 20® 
Mid 70^, fc?r. arc refpcftivcly equal to each other ; 
and confequently the firft great Diviiions from i to 
45^ on the Line Tofarftficial Tangents y will like* 
wife ferve for the C^-Tavgenis of thofe Degrees, that 
is, for all the Tangents from 45^ to 90^, reckon'4 
back again to the beginning of the Line T« And 
this is the Rcafon why on thofe Lines of Tangents^ 
you fee the Numbers placed at each loth Divifion, 
thus iO|8«, 20I70, 30J60, 40|f09 45, attheEnd. 
For otherwife the faid Line of Tangents, muft be 
continued out to double tke Length it now is, which 
would not be near fo convenient. 

12. The Numbers in the 2d and 4th Columns are 
the Divifions on the Gunter^ which correfpond to the 

jfimilar Divifions qn the Lines of Sines and Tangents ; 
wherefore the former being already made, the tw$ 
latter may alfo eafily be conftruftcd by means of that. 
And thefe things are all I judge neceflary to be faid 
here concerning the Conftrudion of thofe excellent 
Lines of artificial Numbers^ Sines j and Tangents \ 
and as to their Ufcs, that will be a Subject to be 
treated of after the fraificalUfe of the Logarithms 
themfelves is firft explained and inculcated j for then 

, the Ufe of thefe Inftruments will be much better con- 
ceived and underftood. 



^ 



M CHAP, 



Si 



'ne Manner ofujmg the 



CHAP. XII. 

(y/A^ Manner £^ufing /£v Tables g^Lo- 
GARiTHMS in rRA.CTiCE \ andoj the Prc7 
xtc^\SiXz% thereto. 

i.fT^HE Logarithms being made for natural 
X Numbers by Ibme of the Methods before- 
going, the next thing neceflary was to difpofe them 
into Tome convenient Order ox Form for praSicalUfes. 
And fuch a Digeji or ColleiSlion of Logarithms, w^ 
call the Logarithmic Canon or ^Tables. 

2. Thefc Tables arc of two Sorts ; the lirft coi\- 
tains the Logarithms of all natural Numbers from U- 
rity or i to loooo, or ipioop (as thofe large Tables 
of Mr. ^herwin.) In thefe, the general Manner or 
Form is fuch as here exprefs'd 
in the Margin, which confifts 
of two Columns ; in the firll are 
placed the Numbers^ in the fe- 
cond the Logarithms eorrefpond- 
ing thereto, with their Indices. 
And three tf thefe double Co- 
lumns fill a Page in common 
Books of this Form 5 and this, 
of all others, is the moft obvious- 
and cafy as to its Ufe, which therefore can need no 
Explanation. For by Infpcftion only is feen what 
Logarithm belongs to' any Number within the Com- 
pafs of the Table. 

3, But tho' the aforcfaid Foriff be the moft natu- 
ral and obvious, yet it is not the moft artful and 
comprehenfive ; therefore another Form or Difpofition 
of the Tables for natural Numbers has been contrived 

more 



Numb. 


Logarithms; 


997 
998 

999 
1000 

lOOI 

looa 
1003 


2.9986951 
2.9991305 

2-9995655 
3.0000000. 

3.0004341 

3.0008677 

30013009 



V 



chap, XIL Tailes of Log AKiTUMS. 83 

more concife, or which takes up le(s room, and is yet 
as perfeA and ufeful as the other ; a Specimen of this 
I^'orm 1 have here anneited. 



N<*. 1 Logarithms. ] 


173 

174 

.'75 


1 X 


2 


3 


4 


238046 
240549 
243038 


238297 
240799 
243286 


238548 
241048 

243534 


238799 
241297 
243782 


239049 
241546 
244030! 



In which the natural Numbers are placed in the Side* 
Column to the Irfi band^ all but the Units Place^ or 
firft Figure of the Numbers, which is found in a pa- 
rallel Column on the top of the Page, in the Order 
*» i» 2, 3 4>^^. as you fee in the Specimen* By 
this means one Column of Numbers fuffices for one 
Page J whereas in the other Form there are three fuck 
Columns, . the whole Page itfelf confiding entirely of 
the Logarithms, which in this Cafe admits of five 
Columns •, but the Indices m here omitted as not be- 
ing ncceflary, fincc they are kiiowln by the Numbers, 
The manner of ufing this Form is yet very eafy. 
i^or Example, to find the Logarithm of the Num- 
ber 1742 ^ againfl 174 in the Side-Column, and under 
the Units Place 2 at the top, I find the Logarithm 
241048, and fince the Number has four Places, the 
Index mufl be 3; where 3.241048 is the Loga- 
rithm compldit for the Number 1742. Thus the 
Lo^rithm of , 17533^3.243782 ; and fo for others. 

4. But the Tables of Logarithms are yet capable of 
a further, ahd much more curious Improvement with 
regard to their Cpntraftion or Concifenels 5 for fince 
the Differences of Logarithms decreafe as the Num- 
bers increafe, 'tis plain thofe will grow very fmail a^ 
thefe become very large j and confequcntly the two 
or tbree firji Figures of the Logarithms to the left: 
will be thtfame for divers large Numbers together in 

M 2 the 



84 



^ Manner (fujing the 



the Canon. Thus for Inftance, the Logarithms of all 
the Numbers between 4168 and 4266, have the two 
firft Figures to the left the lame in every one, viz. 
62. So likewife all the Logarithms between the 
Numbers 9954 and 9977 have their firft three Fi- 
gures the feme, viz. 998, the Difference of the Lo- 
farithms being only in the remaining Figures. The 
'igures of the Logarithms therefore may be reckoned 
of two forts, viz. fuch as are permanent or the iame, 
for certain Intervals \ and iuch as are variable or al- 
ways altering. In this Form of the Canon now\un- 
der Confideration , thefe permanent Figures are 
printed but once for their refpeftive Intervals, and 
that in- the firft Column of Logarithms next the 
Numbers the variable Figures only pertaining to each 
Logarithm make the Subftance, or fill the wholc^ 
Face of each Page, as in the Specimen here fubjoin'd. 



N^ 


Logarithms. 


132 





I 


2 I 3 


4 

1888 


12.0574 


0963 


1231I1560 


133 


3852 


4178 


45044830 


5^5^ 


134 


- 7105 


7429 


7752 8076 


8399 


135 


13-0334 


0^55 


0977 


1298 


1619 


136 


3539 


3858 


4177 


4496 


4814 


^37 


6721 


7037 


7354 


7670 


7987 


i;^{^ 


9879 


♦0194 


0508 


0822 


1136 


^39 

945 


14-3015 


3327 
478 


3639 
524 


3951 
570 


4263 
616 


975^3^ 


946 


,891 


937 


983 


*029 


075 


947 


976.350 


39^ 


442 


487 


533 


948 


808 


854 


900 


946 


991 


949 


^yy.266 


312 


■35^ 


403 


449 



. 5. The Numbers here, as in the laft Rfrm^ are, 
for the three firft Figures to the left, found in the 

Side- 



Chap. XII. J'ahUs ^/Xogarichms. 85 

Side- Column, the other Figure at top. ThcLogft' 
fkhms in this Specimen are 65, yet but fix of thwn 
need be cxpreffed at length, viz. thofe for the Num- 
bers 1320, 1350, 1390, 9450, 9470, 9490. The 
Logarithm for 1320 is thus wrote, 12.0574, to de- 
note the two firft Figures 12 (feparated by a Dot) 
are permanent thro* the Interval between 1320 and 
J 350, that is, they belong to the Logarithms of all 
the internaediate Numbers between thofe two, and 
therefore need be exprefs*d only for the firft y the 
other Part of the faid Logarithm which is variable, 
is exprefs'd alone in all the reft. ., The upper Part of 
this Specimen or Tariff conC^ of Logarithms having 
two Figures in the permanent Part i the lower part 
is an Example of Logarithms having three Figures 
in the permanent Part. To find a logarithm there- 
fore to any given Number, will alio be very eafy ia 
this Form. Thus, fuppofe the Logarithm be fought 
. for in the Number 1323, 'tis found in this Manner* 
Take- the permanent Part either againji or next above 
the three firft Figures 132, which here is 12, then 
againft 132 and under 3 at the top, you find thex;^- 
ri able Part 1560, to which prefix tht permanent Part 
12, and you have the Logarithm 121 560, which 
with the Index J is 3. 12 1560. Again, to find the 
Logarithm of 1374, take 1 3 the permanent Part next 
above the three firft Figures 137, then againft 137^ 
and under 4 at top, you find the variable Part 7987, 
which annexed to the former Part 13, make 137987, 
and with the Index^ 3*^379^7% ^^^ Logarithm 
fought. 

6. Thus alfo you proceed when the permanent 
Part confifteth ot three Figures^ where the Intervals 
are much ftiorter. One caution only is neccflary, 
and that is, that you obfervc ih thofe Lines where 
there is found an Jfterifm^^ to join all the pariabli- 
Parts in that Line before the *, to the permanent 
Part next above 5 and all after it, to that next ^elow.^ 

'* Thui 



86 tbelianfikr ofujtk^ ite 

Thu^ in Che 7th Line^ aiid id Column, )^ou feeiri 
jifterifm ♦, therefore the variable Part ^Zj^ beforq 
it muft be joinM to r^ tht permanent Part next above 
it, to form the Logarithm 3.13918751 for the Num- 
ber 1380: ^Mt tht variable Parts 61^^ 6568, &ff. 
following it^ arc to be annexed to 14 the permanent 
Part next below^ to form the Logarithms 3. 146 194, 
3.140508, tfr. for the Numbers 1381, 1382, (^c. 
The Reafon of which I prefume muft h^felf -evident 
to every Reader. 

7. I have Been ihe more prolix on this l(fi Forfk^ 
left any Obfcurity or Uncetiainty (hould remain to 
difeourage or ptejudicfe tht young Tjro againft fb rari 
and fo advantageous a Contrivance. I fay, fo rare ; 
becauie I have never fcfett (amongft many) above ohe, 
viz. Sberwitfs Canon, in this Form ; and that; by 
reaibn of its great Price^ is not very common; Tho* 
that Gentleman (ays in his Preface, he has ih this ex- 
cellent Method followed Dr. John Newton in his 2r/- . 
gonometriaBritannicaj a Book which I have notfeen. 
The Advantage alfo of this Abbreviation is next to 
that of the Invention it fclf ; for hereby the prolix 
and unwieldy Tables (in their original Form) are re- 
duced or abridged to one half the Bulk nearly ; all 
thefuperfluous Part being omitted, and nothing but . 
what was ncccflary retained in this Canon. 

8. According to this moft excellent Abridgment 
therefore, I have firft of all, thlt I know ibf, pub- 
lifhed the common. Canon of Logarithms forNum* 
bers from i to loooo ; having taken the no fmall- 
Pains of tranfcribing the whole with my own hand 
from the aforementioned large Work of Mr. Hen. 
Sherwinj which is the moft correft of any extant. 

9. In this Form or DifpofitJon of the Canon, I 
have alfo publifhed the Logarithms of Siiies and Tan- 
gents \ which thing hath not been done before in any 
Work great or fmall," that I have ever feen or heard 
©f. This makes the 2d Part of the Logarithmic 

Tables, 



Chap. XII. Tables ^/Logarithms. 87 

Tables^ as mentioned Art. 2. And fince they are 
here in adifFerent Form from all others, it may not 
b^ unn^freflary to hint to the young Learner, that 
^he Numbers exprefs'd in the Side-Column are the 
Degrees, and every xoth Minute, and the Numbers 
in the parallel Column at top are the Minutes between 
the loths^ fee the following TariiF of the Logarithms 
of Sines in this Form for the Minute^ from 72 De- 
grees to 73. 



"dT" 



72. o 
10 

20 

40 

59 



978.206 

615 
979.019 
420 
816 

980.208 



3 4 



247288329I370 

655\^96 7 3^.777 
059 100 140 180 

+60500539579 

« 55^5 934-97 3 
247l286J325,364 



'!('o give an Example $ ^t the Logarithm be fought 
for the Sine of 72^ 43^ Seek in the Side-Columi^ 
72^ 40', next* above which is the permanent Part of 
the Lc^arithm 979 in the firft Column of Loga«- 
rithms ; then a^inft 72^ 40, and under 3^ at top 
you find the variable Part 934, whic^ annexed to 
the other makes 979934 ; to which prefix the Index 
(which is fet at the top of each Page) 9, and the Lo- 
garithm is compleat, viz. 9.979534 for the Sine of 
72° 43' » and thus you proceed for any other. 

10. I have contracted the Logsfit^ms to iix Places 
of Figure; 6n]y, as being fufficient in. common Ufe -» 
th? natural Sines and Tangents ar? not h^re infertedi^ 
lince when their Logarithnis can be ufed, they tJicm- 
^elves are ufelefs. Befides, whenever they are re- 
quired, they may be imrnediatcly had from their 
Logarithms. For Exan^ple, fuppofe X would know, 
the natural Sine zt^d Tangent for ^S^ 47^, the Loga- 
rithm, Sine and Tangent of this Arch, arc 9.796836, 

and 



S8 The Manner rf ufing the 

and 9.965609; To fchcfe Logarithms (neglcfting their 
Indices) find the natural Numbers, by the firft Part 
pf the Canon, they will be 626377 ^^^ 803542, the 
natural Sine and Tangent fought. (See the Method 
below. Art. 15. for ftidmg the Number of a giveii 
Logarithm.) 

11. The Logarithm of the Secafis of any Arch, as 
of j8®47', is thus eafily obtained : 

From the double Radius 20.000000 

Subftraft the Co-Sine of 38^ 47' 5= . , 9.891^27 
There remains the Log.Secant of 38^47'= 10.108 1 73 

And thus the Logarithm of any other Secant may 
be fquod, and confequcntly the natural Secant^ }ox 
natural Number belonging thereto. 

12. The Reafon why the Indices^ on tjie top of 
the Pages, of the Logarithms of Sines and Tangents^ 
are ib large, viz. 7, 8, 9, 10, 1 1, fc?^. is becaufe 
the Radius of the Circle was fuppofcd to confift of 
1 0000000000 equal Parts, whofe Logarithm there- 
fore is lO.oooopo ; wherefore a Number of fuch e- 
qual P<?r/Jexprcfliiigthe Sine of one Minute i', will 
confift of 7 Pieces, whofe Logarithm then ^ill have 
its Index 6. The other Sines will confift of 8; 9, and 
10 Places, and fo the Indices of their Logarithms 
will be 7, 8, 9, as in the Tables ; thus alfo the 
Numbers expreffing'the Tangents, in fuch equal Parts 
will confift of 7, 8, 9, 10, It, 12, 13, and 14 
Places of Figures, whence the Indices of their Loga- 
rithms muft be 6, 7, 8, 9, 10, II, 12, 13, accord- 
ing to Art. nth and 12th of Chap. I. But firice, 
as before faid, the firft fix Places of the Logarithms 
to the left are fufBctent, the reft are rejefted as fu- 
perfluous. ^ 

13. In future Operations there will be frequent 
Occafion for what is called the Arithmetical Complex 
ment of a Logarithm, which is nothing but the I)//- 

ference ' 



Chap. XII. Tables of Logarithms^. 80 

ference between that Logarithm and Logarithm-Ra- 
dius id.oooood. 

Thus if from 10.000006 

You fubduftthe Logarithm 4.877026 

There will remain the Arithmet. Comp. = 5.122974 
And this is done mentally in an Inftant, by taking 
every Figure from 9, except the fifft, which you take 
frorii io; 

Note. If the Logarithm be of any Sine or Ian- 
gent, add 10 to tht, Index of .the Arithmetical Com-* 
plement, and it will be the Logarithm of the Co^ 
Secant of the fame Arch. For Example, 

Suppofe the Log; Sine of 3^ 48' = . . . 8.821342 
The Arithmetical Comp. thereof is . , . i . 1 78658I 

to which add 4 . . . ^ 4 . . i lO, 

The Sum is the Log. Secant of 8^° 1 2' = 11. 178658 

Which is evidently the fame Operation as that in 
Art. II. hereof. 

14. From the theory oi Logarithms we learn, tbaf 
the Differences of great Numbers are proportional to 
the Differences of their Logarithms. (See Chap. V. 
Aft. 6, 7.) Therefore tho* the Canon of Logarithms 
gdes no farther than the Number 1 0000, it may by 
this means be extended to the Number 1 0000000, 
or the Logarithm of any Number under 1 0000000 
may be found by the prefent Canon^ according to 
the following Rule. 

Firft ; find the Logarithm of the four firjl Figures 
6f the given Number, by the firft Part of the Canon. 

Secondly ; fubftraft this Logarithm from the Lo- 
garithm next greater or neS!t following in the Table j 
and referve the Difference. 

Thirdly ; multiply the Difference^ by the remain^ 
ing Pigiires of the given Number 5 and from the Pro-- 
du£l cut ofF to the Right handy fo many Figures as 
there werfe remaining in the givtn Number. 

N Fourthly' i 



"1 



9© 7he Manner ofufing the 

Fourthly ; add the Remainder of the ProduSl to 
the Logarithm firft found, the Sum ihall' be the Lo-* 
garithm fought. 

For Example, let the Logarithm of the Number 
127053 be fought. 

The Logarithm of the t ^^^ o 

• firft 4 Figures } ' • "7^00 = 5.103804 

The next greater Log. isof . . 127100 = 5.10414^ 

The Differences 100 342 

Wherefore fay, as 100: 342 :: 53 : 181,26 

53 

1026 
1710 



181I26 

Add the firft Logarithm 5.io;?8o4 
The Sum is 5.103985 '= the Loga- 
rithm fought for the Number 127053. 

Example 2. Required the Logarithm of the Nuxn- 
bet 3567894? 

The Log. next following is of 3^68000 = 6.55242 5 
The DifF. of Numb, and Log. 1 000 122 

Th^n fay, as 1000 : 122 :: 894: 109,068 
Add the firft Logarithm . .6.552303 
theSumisthe Log. 3567894=6.552412, as was re- 

quired. And thus you proceed for the Logarithm of 
any other greater Number than thofe in the Canon. 

15. By a Method reverfe to the foregoing, you 
find the Number correfponding to a given Logarithm% 
thus, fuppofe the given Logarithm be 3.567026, 
and you would know the Number thereof. Seek 
this Logarithm in the Table, and becaufe you there 
find it exaSlly^ the Number 3690 correfponding 
thereto, is the Number fought. But if the given 

Logarithm 



i 



Ghap.XII, Tables g/' Logarithms. 9! 

Logarithm be noUxaSily contained in the Table, and 
more than 4 Figures be required, proceed as follows. 

Firfk ; feek in the Table a Logarithm the next kfs 
to the given one, for the four firft Figures of the 
Number fought. 

Secondly \ fubftraft this Logarithm from tht given 
cne^ and annex to the Remainder, ^^ many Qypbers as 
you feek Figures more than four. 

Thirdly ; take the Difference between the Loga- 
rithm juft found and the next greater^ by which di- 
vide the faid augmented Remainder ^ the ^otient an- 
nexed to the four firft Figures ftiall compleat the 
Number fought. 

Example i. Let there be fought theNumber to the 
Logarithm 5.103985, to fix Places of Figures. 
The Log, nextlefs is of . . . 127000 = 5.103804 
The given Logarithm . . . , 5.103985 
The Difference or Remainder 181 

The next greater Logarithm 1 2 7 1 00 = 5.104146 
DifF. between the greatefi and leafi Log. 342 

Since the Places iu the Number fought are 6, aug- 
ment the firft Remainder 181 with two Cyphers^ and 
it will be 1 8 160 i then 342)i8ioo(=53, which an- 
nexed to the four Figures 1270 before found, make 
the Number 127053 required. 
Thus alfo to the given Logarithm 6,55241 2, you may 
find its proper Number 3567894, and foibr others. 

16. In the fame manner you. proceed to find any 
Pecimal Number to a Logarithm given, only in this 
Cafe the Indices of the Logarithms are negleded rill 
the Operation is finifhed, and then fo many Figures 
are to be cut off from the Number found for Deci- 
mals, as the Index of the given Logarithm Ihall indi- 
cate ; what is here faid relates to plain or termi-- ' 
nate Decimals only ; but there are other forts of De- 
<:imals which circulate or perpetually repeat one or 

N 2 xpore 



'94 ^e Manner ofujing the 

more Figures ad infinitum : Andthofe Figures which 
thus circulate zrt Repetendsj as in thefe Numbers, 
viz. 235,2222, £5?r. 27,83333, ^c. 2.383838, &c. 
702,6026026, fcff. 0,2672326723, fcfr. Now thefe 
Repetends need be wrote but once if we flur the firfli 
and laft Figures in each, to denote them fuch, as 
thuS235,A 5 27,85- ; 2,?* ; 70^,60 ; 0,;2;6725', 6ff- 
See more in my Univerfal Syfiem, or Body of Decimal 
jiritbmetic^ printed for Mr. Noon. 

ly. The Logarithms for the Re- [" ^=0.045757 
peating 9 l^igits arc made by adding ^=0.346787 
the Arithmetical Complement of the 5'=i:o.522879 
Logarithm of 9, to the Logarithms 4=0.647817 
of the faid Digits, and are fuch as< ^=0.744727 



^=0.823909 

7=0.890855 

«=o.948847 

L ^=1.000000 



here annexed. The Lqgarithms of 
pure compound Repetends are ipade by 
adding the Arithmetical Complement 
oifo many ^\ as there are Figures in 

the Repetendi to the Logarithms of thofe Numbers 
confidered as terminate. Thus the Logarithm of the 
Repetend 56, f is found as follows : 

To the Logarithm of 36.5=1.562293 

Add the Arith. Complement of . . 999 =0.000434 
The Sum is the Log. of the Repetend 3'6,f=i. 562727 
and thus proceed for others. 

;8. If the Repetend have any prefixed terminate 
Part ; then from fuch a mixed Repetend fubftraft its 
terminate Part^ and to the Logarithm of the Re- 
mainder add the Arithmetical Complement of tbp 
Logarithm of as many 9's as there are Figures in the 
Repetend. For Example, fuppofe you would fina 
the Logarithm of tht mix*d Kepetend 26S'92,7, pro- 
ceed thus; 



Chap.XIL Tables (>/ Logarithms. ^^ 

From the Repetend 26892,7 

Subftradl the terminate Part 26 

To the Log of the Rem. . . 26890,1=4.429592 
add the Arithm. Complem. of 9999 =0.000043 
The Sum is the Logarithm of 2 6«'92, 7=4.429635 

JVJ7/1?, the Indices of the Arithmetical Complements 
are here (as in thefe Cafes they always muft be) o- 
mitted. 



CHAP. XIIL 

Of the Oxigmand Conftruaion g/'SHAKERLY*i 
^«^Street'5 Logistical Logarithms. 

^*. fTT^H E Ufe of Logiftical Logarithms is in AJlro^ 
X nomical Calculations^ or Sexagejimal Arith^ 
metic •, but this fort of Arithmetic, which taught the 
Rules of Addition^ SubJiraSiionj Multiplication^ Di- 
vifiony &c. of Semgefimal pra£lionSy viz. Degrees^ 
Minutes^ and Seconds of Motion or ^ime^ was, xafor^ 
mer timesj called Logiftical Arithmetic, And fince 
the Invention of the cornf^on Canon of Logarithms^ 
Mr. Jeremiah Shakerly^ in his TabuU Britannica^ 
firft contrived from them a fort of Logarithms ad- 
apted to the Rules of Logiftical Arithmetic ; and 
therefore gave them the Name of Logiftical Loga^ 
rithms. And fince him Mr. Thomas Street^ in hb 
Aftronomia Carolina^ has invented another and more 
convenient Form oi Logiftical Logarithms ^ than Sha- 
kerly^s. And fince Tables of both thefe forts of Lo^ 
giftical Logarithms are extant, 'tis proper to acquaiint 
the Reader with the ConftruSion of both, which is as 
follows. 

2, 



$4 Of Shakcfly'i and StrcctV 

2. Since Logiftical Logarithms are altogether con# 
cerned in working Proportions of Degrees^ Minutes^ 
and Seconds^ and more efpecially of Minutes and Se^ 
£onds^ t<^ether with Integers^ *tis evident, if thofe 
Sexagefimal Fra6Hons were reduced into the loweji 
J)enominationj viz. of Seconds ^boc. they might then 
be work'd with the Logarithms of common Numbers. 
Thus fuppofe the Proportion be 6d oo^ : 3' 47'' :: 
51^ 29^ : 3' 15^5 if thefe/r^<^/^»^j/ Numbers be re- 
duced to Seconds y they will ftand thus, 3600^ .-227^ 
:: 3089^ : 195'' ; wherefore *tis plain, the Propor- 
tion in this Cafe mfiy be wrought by the common Ca^ 
non of Logarithms^ as will be hereafter fliewn. But 
then as here is no Radius^ there will arife double 
Trouble in the Work by Logarithms in common Ufe, 
in firft adding the Logarithms of the two middle 
Terms^ and then fubJiraSing the Logarithm of the 
^rji from that Sum^ in order %to have the Logarithm 
of the fourth Term fought ; or elfe the Complement 
Arithmetical oi thtljyg2Lx\\}\m of X}citfirji Termmuik 
be taken to perform all by Addition only. To avoid 
therefore the Trouble attending perpetually either of 
thefe Methods, 

3. Mr. Shakerly makes this Proportion, as 3600^ : 
227''^ :: 1 00000, ^c. : 0.06305, fc?r, or with the 
Logarithms, thus 5 

As the Logarithm of 3600'' s= 3.556302 

to the Logarithm of 227''' = 2.356025 

fo theLogarithmof Radius i.ooooo,&c.=io.oooooQ 

to the I^ogarithm of o.o6305>&c.= .8.799723 

Now 'tis plain, the two lafl Logarithms perform the 
feme as the two firft ^ their Properties being the feme ; 
but the firft of the two latter Logarithms is Radius 
lO.ooooQO, which therefore he called the Logifticat 
Logarithm of 3600''' or one Degree, and confequently 
the Logarithm .8.799723 is the Logiftical Logarithm 
of 2 2 yyz^' ^f. And thus the Logifiical Logarithm 

of 



Chap. XIII. i^i^/w/LoGARITHMS. iJ5 

of any other Degrees^ Minutes and Seconds may be 
found ; viz. by reducing them to Seconds^ and then 
by taking from the Logarithm of thofe Seconds the 
conftant Logarithm 3.556302 of one Degree in 5^- 
€onds. 

4. Thus from theLog. of. . . I'zzSo^-rzi.y^j^i^i 

Take the Log. of one Degree =3600^^ =3.556302 

therercmains theLogift. Log. of i'=6o''=*8.22i849 



i^ 3' 00^=3780^= 3.57749^ 



Again, from the I 
Log. of y 

tzke the conftant Log. 1^ =3600^^ = 3.556302 

X^riTof""} '" 3' oo'=378o'=.o.o.„so 

By thefe Examples you eafily perceive how the Lo- 
gtftical Logarithm may be found for any Number of 
Degrees J Minutes and Seconds^ in Shakerlfs Form. 
Andfince there are 60' or 3600^ in^^ Hour^ as well 
as in one Degree^ therefore a Table of thefe Jjogiftical 
Logarithms ierves equally as well in the Computation 
of ^ime as Motion. 

5. Butif theTimeofa whole 
Day or 24 Hours be the Inte- 
ger^ fince there are but 1440' 
therein j and 3600^': 1440' :: 
2'^i: i'; alfo fince 60' : 24HO 
l:2'i : iH^ ; therefore if, thro* 
the courfe of thefe Tables of 
Logiftical Logarithms^ againft 
every I'i you place the Hours ^ 
and againft every 2^'i (or rather 
every 3d and 5th Second) you 
place the Minutes of an Hour ; 
the Table of Logiftical Loga- 
rithms ioT Motion andTiW will 
be compleated ; a Spedmen of 
ivhich, in this Form of Mr. Sba-^ 

kerky% 



° 'Motion 


Time. 


II 4^ 


H . 


H 


I 
2 

3 
4 

5 
6 

7 
8 

9 
10 


2700 

987506 
987522 

9'^7SZ'^ 
9^7554- 
987570 
987586 
987602 
987618 
987634 
987651 
gSy66y 


XVIII. 
. _ _-_ 

I 

2 

3 

4 



'p6 O/^Shakerly'j and Street*^ 

kerlef Sj I have before annexed. The firft Cdliiniti 
contains the Degrees, Minutes or Seconds -^ the fe- 
cond Column the Logiftical Logarithms thereof; atld 
the third Column contains the Minutes of Time, the 
Hour being exprefs'd at the top, viz. XVIII. aft- 
fwering to the Motion of 45', or 2700'''. 

6. But becaufe thefe Logiftical Logatitbms of Mr, 
Sbakefley, confift of many Figures throughout the 
Table, it minifter'd occafion to Mr. 7*homas Street to 
contrive a more compendious and convenient Form 
of thefe Logarithms ; and fuch he invented, which 
tho* large at the beginning of the Tabk, yet imme- 
diately leflen very faft, and fo continue to the End 
of the Table, or 6d oi^6o& ; whofe Logiftical La^ 
garitbm is =0. / . 

7. The Reafon of "vfrhich is manifeffi from tKfe 
Manner of their Conftruilion, which is as follows'; 
Suppofe any Proportion of Sexagejimal Numbers^ as 
that before made ufe of. Aft. 2. viz. 60' : 3^47'^ :: 
51' 29^ : 3' 15^, whidh reduced to Seconds, ftands 
thus, 3600^' : 227''':: 3689-^: 195^^ Now in ordet 
to obtain a vacant ^erm in this Analogy, Mr, Street 
(inftead of Sbakerly^s Analogy 3600'^ :22y^::i 00000 : 
6fr.) inverts the Jirft Ratio, as thus ; 227^^^ : 3600'^ 
:: Unity ; to a fourth Number whofe Logarithm is 
reputed the Logiftical Logarithm of the firft Term 
227'^^, as the Logarithm of Unity is ofthefecond 
3600^^^ See the Work. 

The Logarithm of ... . 3' 47''^=227'''=2. 356026 
The Logarithm of ; . i°=:6o'=z^6oo^=3.556302 

The Logarithm of Unity 0.000000 

The Logift* Log. of . . . 3' 47^''=2 2 7^^=^1200276 

8. Whence it evidently appears, that to find the 
Logiftical Logarithm of any Number of Seconds, yoU 
need only fubftrad the common Logarithm of the 
Number of Seconds from the conftant common Lo- 
garithm of 3600*, for the Remainder or Difference 

will 



Chap. XIIL ?2iiJ5pi5f Logarithms. 97 

Wifl be Ac ZtgiJtkalJL^ariibm requiitd, in S^e$^9 
Fomi. 

Thua from the Logarithm of, • 3600*^:2:3.556302 
Subftrai^ the Log. of £t' 2^''=3o89 =3.489818 

There remai» the Logift. Log, of 5 1' 29^= =^. ,66481 
Agun from the &id Log. of. . 3600^=3. 5563 q^ 
Subdufl: the Logarithm of 3' 25^=195=2.290035 

There remains the Logift.Log. of 3' ts^^slziSzij 

9. From wheace *tis evident, that the greater the 
Kumber of Seconds is, the lefs will be the I^x^f- 
tjcal Logarithm thereof ( till you come to the Num- 
ber 3600^ whofeLogiftical Logarithm is nothing at 
«1], gs before faid. And thus it appears that the Ixk 
g^ithmA of this kind of Mr. Sireei^s Form, have la 
them fewer Places of Figures, and are therefore more 
convenient forUfe, by much^ than thofc of Sbakerly^s 
Form, before defcrib'd. And for that Reafon I have 
4chofe to give the Reader a Table of Street*^ Loga- 
jithmsmtherthan the other; and tho* Mv.Leadbetter 
has gjyen us Tables of both forts, yet I think it in*, 
tirely needle& ^ fince all the principal Uies of ^ibif^ 
hrfy% are piuch better performed in Streets LogiA 
tical Logari^ms. 

10. In fhewjng the Manner of making thefe Lo- 
garithms from the common Ones, I have exprefled 
tnematiarge,v/2. 1200276, 66484,1266267, the 
Log^ftical Logarithms of 3^47^ 51' 29^, and 3' 15^, 
as p^r Art 7, 8, Yet h^re two things are to he ob- 
ierv*d: Firft, that the Index is not diftinguifticd 
from Logaritfim itfelf, with a Pointy as in the com- 
mon ibrt I but the remaining Figures, both of Lo- 
g^thms and Indices^ be they more or lefs, are re- 
puted toge^ther, the Lpgiftical Logarithm. Secondly, . 
that two Places of Figures to the Right Hand in the 
Examples, are ftruck offin theTabfe ; the other be- 
ing fully, fufficient for all the Purpofes thereof. So 

O that 



9? O/'Shakerly'i and StrcctV 

that m the Table you will find the &id Logifiical Lo- 
garithms vfwtt 12003, 665^ 12663, &c. 

11. Mr. Leadbetter has taken the pains to continue 
the Table of Logiftical Logarithms in Street's Form, 
to i2o\ or 2 Degrees ; but as there is little occafion 
for any more than the Logiftical Logarithms of 60' 
or I Degree ; and when there is, the fame Logarithms 
are capable of anfwering it, 1 have therefore con- 
tinued them no farther than the Inventor did, viz. 
to 60' or 3600^ 

12. As to the Form of the Table, •tis very cafy 
to be underftood, efpecially if thofe of the common 
Logarithms of Sines and Tangents before defcribed 
are : The firft Column of the Tables contains theD^- 
grees or Minutes^ or Minutes and Seconds in Sexa^ 
gejimals^ as in the common Tables^ in the Order of 
Denaries or lo's, the nine Digits running along on 
the top of the Table, under which, in the; feveral 
Columns, are the Logiftical Logarithms abbreviated 
in the fame manner as thofe of Sines and Tangents ; 
and are to be taken out according to the Direftions 
there given, which fee. In the laft Column are con- 
tained the Numbers of the firft reduced to Minutes 
or Seconds ; and are to be compleated likewiie with 
the Digits on the top of the Table. Thefe are re- 
ferred to when the Logiftical Logarithm of any inte- 
gral Number is fought. An Example or two will 
render all eafy. 

13. Let it be required to find tht Logiftical Loga^ 
rithm of 2^ 48' or 2' 48^ Firft feek 2 40 in the 
firft Column, and agaidft it and under 8 at top, you 
find 310, which annexed to its proper permanent Part 
13 in the firft Column of Logarithms makes 133 10, 
the Logiftical Logarithm of 2 48, as required. Thus 
the Logiftical Logarithm of 37' 59^ is found to be 
1986 5 and of 54' 40'' to be 404 5 and of 58' 37^ to 
be loi 5 and of 59' sf to be 8. And thus the Z.^- 

gijlical 



Chap.Xm. Talks of Log AKiTKMS. gg 

giftical Logarithm of 60', is =0 ; and fo will be a va- 
cant Term in all Analogies for Operation. 

14. Let it be requit^ to find the Logiftical Lo- 
carithm of the Number 584. Firft feek 580 in the 
kft or right-hand Column, and againft it and under 
4 at top you fee *99, which ftiews it muft be join'd to 
the following permanent Part y% in the firft Column 
of Logarithms, and therewith makes 7899, the. Lo- 
garithm fought. Thus the Logiftical Logarithm ^ of 
1000, is found 5563 i and of 33599 to be 301 ; and 
of 3596, to be5 -, and that of 3600 is nothing. 

15. This Table alfo equally ferves for Time, whe- 
ther for z Day z,nd Minutes^ or Minutes ^nd Seconds -^ 
by help of the little Table at the End of the Lo^ 
giftical Logarithms J which fhews what Parts of Mo- 
tion in Degrees and Minutes correfpond to Time in 
Hours znd Minutes. Thusagainft 13^ in Time, you 
fee 33' of amotion j againft i Hour is 2^ 30' ; and the 
Motion aiifweringto VIP 43', is 17^ 30', -f" ^^ 4^' 
= 19^ 18' 5 and that anfwering XXP 21' is 52^ 30V 
4. o^ 53' = 53^ 23' ; and fo for other Parts of Time j 
confequently * 




The Reafonof all which is evident, from Art. 5. 
hereof. 

16. Thefe Logiftical Logarithms may in like man- 
ner be rendered applicable to Computations of Money ^ 
Weights, Meafures, &fc. Thusfmce 60^ : 20» :: 3 : i, 
therefore the Shillings in a Pound correfpond to each 
3d Degree or Minute oi Motion. And again, fince 
gee*" : 3600'' :: i : 3'i; thereforeto i, 2, 3, 4, fcfr. 
Farthings there anfwcrs 3', 7', li', 15^ &?^. Minutes 

O 2 of 



too Of Shtkw\y*sand^ett^s fables, Scx^. 

of Motion h whence a Table auiy be formM to fhew 
the Lo^iilical Logarkhm ^r any Number of Far* 
things under 960^ or one Potind* And thus yoK 
may proceed to frame a Tabic to render thefe Logif- 
f teal Logarithms for Motion useful for the Ounces snd 
^unds in an hundred i9^eigh 1 and thtt with great 
Eafe^ fince 1792, the Ntimber of Ou«ce$ in zn hun- 
dred Weight Averdupoife, a. nearly half the Numbcar 
3600^ the Seconds of the Table. ConfequeAtly^ 
the Logiftieal Logarithm of the Dwble of any Num- 
ber of Ounces, is that requited for the Pmnds au4 
Ounces eq^ual thereto. 

17. Or, kftly, tis caf^ for any one who fliall 
think it worth while, to calc?u!atc Tables of Lpgif- 
tical Logarithms^ for any Species of Con\putation» 
peculiar to it felf. Thus with refpcft to Money, if 
the common Logarithms of all the Farthings under 
96a, be fubdufked from the conftant Logarithm of 
960^ the Remainders wiH be the Logifticat Loga- 
' ifithms of the Farthings in a Pound. The common 
Logarithms of all the Nwnibers of Ounces under 1 792 
fubftra<5ted from the conftant L-ogarkhm of the faid 
Number 1792* will kave the l^gifiical Logarithms 
of all the Ounces in an hundred Weight. And thus 
you proceed to make Logifiical Logarithms of any 
Kind you plcafe, or may have occaiion for. Which 
is too eafy a matter to require an Exampfe* befidea 
thpfe above,. Art. 7, and 8. . 



if3m 



LO- 




LOGARITHMOLOGY. 



PART IL 



i.,i-.w»ipa -t - ■ I ■ i*^ '" "''^' 



TZ^Praxis ©/'Logarithms, Common 

and Logifiical : 

• ' 
With //J AppLicATio«r to Fuhar and Duo^ 
decimalJrithmetiCy Plain anaSpbericalTri^ 
gonomefry^ Navigation^ Menfuration ofSuper-- 
jicies and Solids y Gauging^ Timbtr-'Meafure^ 
Agronomy ^ perform" d Numerically andlnjiru^ 
mentally. 



ClfeclJSiilJ&fed^^ 



CHAP. L 

Cftbe Rules ofADDiTion^ Scjbstraction, 
Multiplication, and Division of the 

laciicCS ^XoOARITHMS. 

/\ prcmifed, the Ruks for a PraSical U/e and 
JljL Management of thofe artijkial Numbers 
vill from thence be eafily underftood, and the Ra^ 
tionale of every Operation be apparent to the intelli- 
gent Reader. Before we proceed to the Ufe of liO- 
cajithois as aralied to common Arithmetics &c. wis 
^ ' ^ ' muft 



102 Addition of the 

jnuft firft confider {omt previous Rules and Methods 
which regard the due ordering and working thofe 
Nqn^bers themfelvesy on account of the Indices, 
which admit of divers particukr Cafes in the Rules of 
4ddition^ SubJiraSiony Multiplication^ and Divijion : 
which therefore muft be exentplified and illujlratedzs 
in the Sequel of this Chapter, 

Addition ^/Looarithmst. 

2. As in all Species of Arithmetic the ^r&funda^ 
mental Rule is Addition^ fo in this of Logarithms, 
the Rules which require ^\s primary Operation ooxoc 
firft to be coniiderM^ and they are as follow^ 

{Note^ I ftiall in this Pls^ce call the Indices of Loga- 
rithms of wA^/^ iV«/»^^r/, Integral Indices i and 
thofe of the Logarirhn^s of Decimal or Fra£ito^ 
nal Numbers, Decimal Indices.) 

Ruk I. If the Indices be htb Jnlegralj add them 
together for the Sum required. 

Rule IL If the Indices are hothDecimal^ add them 
as before ; and obferve, (i.) if the Sum be above or 
juji ip or ICO (=tA ; fee Chap. IV. Theory) caft 
away lo or loo. (2.) If the Sum be under 10 or 
100 (=tA) add 10 or 100 thereto j and both the 
Sum in the latter Cafe and the Remainder in tht for- 
mer^ will be Decimal, 

Rule III. If the Indices be of diffkrent Kinds, vi%. 
Integral with Decimal^ the Sum^ if under 10 or 100, 
is Decimal \ if juJl 10 or 100, or above^ caft away 
10 or 100, the Remainder is Integral. 

Rule IV. In cafe the Sum of two or more decimal 
Indices be Jefs than io=tA, the beft way will be lo 
ufc the larger decimal Indices y where tA=ioo ; and 
then their Sum will be greater than 100, and fo the 
Reafon of the Operation will be more evident. 

3. All tbefe Rules re^te to the Indices of the Lo- 
garithms only, arid are exempjified as/ollow : 

Ex- 



Chap. I. J«<//V« ef Logarithms. 103 

Examp.I. To 3-513217 Exam. II. 2.317227 
add 2.303196 . 0.850891 

Rule I. Sum = 5-816413 Sum = 3^168118 

Exam. IIL To .9.849235 Exam. IV. .97.237406 
add .7.786822 .95.072607 

^^L ] Sum= .7-636057 Sum = .92.310013 

Exam. V. To .4.273760 Exam. VI. .62.346174 
add .3.067247 .21.300725 

% ".■ } S. = .17-341007 Sum =.183^646899 

Exam.VII. To. 6.372458 Exam. VIII. .88.426703 
add 2.673842 5.268402 

Rule III.Sum=.9.04630o Sum = .93.695105 

Exam. IX. To 5.206737 Exam. X. 8.426735 
add .8.312046 .92.105374 

Rule III. Sum=3.5i8783 Sum= 0.532109 



*d*i $ .2.070346 
II. 

Part z. 



Rule II. ] „ „, o 

,.J S. = . 27.586181 



r Or rather thus, per Rub IV. 

Ex.XIi. .94.203106 

.91.3127^-9 
.92.070346 

.77.586181 



»Ti8 poffibk thefetwo laft Examples tnayfeem fome- 
what oifcure •, but the Reafen and 7'rutb of each will 
appear, if they are wrought at twice, as (oOfrfrs : 
» To 



104 Addition of the 

To .4.203106 And -.94.203105 

add .1.312729 .91.312729 

Part a. J S. = .15.515835 Parti. J S.= .85.515835 
to which add .2.070346 .9*2.070346 

Adell. 1 ^ -7-— Fule II. \ ^ 

iyt».XS. =.27.586181 Part I. J S. = . 77.58618 1 

' ■ H I ' I 

FiXMn wbcAce •appears the Reaiba why» in thefe 
Cafes, the larier Indicts are preferable to thtfrnaller 
mus. 

SuBsra ACT ION i^ Looa r i thmts. 

^ Tn thfs SuhJlraSm of /;^^V^j the following 
RuUs are to be obiervM, viz. 

Rule!. If they are both J}i/irj^/, and the higher 
one ^ greater^ tYit Remainder m\\}U Integral. But if 
lbs i(Mtfr one be the gre^ier^ then add lOto the higher 
oiie« md j|i#rtfff ^ the Remaini$r will be DerimaL 

Ruk U. If both Indices are Ikeimal \ and the 
iBi^i&^ be the greater 5 the Renuiinderh Integral ; if 
not, add 10 or ico to the i&if*er one, and iiibduft 
the lower^ the Remainder mil be Decimat. 

Rule ni. If the Indices are pf different ibrts, ^va:. 
one Integral the other Decitnai j then if the Af^iiw^ be 
Integral^ add 10 or 160 to it, and iubftraft 1 the 
Rmmnder wUl be Ii^^gral. But if the i^ig'i&f r be D^* 
fiwr^ aod thci/fM^^r, the R^m^indfr will be Dra* 
nufl ; illeffer^ the /^wy^r /|Mf^/^ muft be ufed. 

5* The Rules alfo ate eafily dmy'd fipm the fore^ 
mtng fbe^^ and are ill«^iattd by the fpUowing 
£xan4>Ies. 

Exam.I.From 5*8i64i3 Ex. U. 3.168118 

Suhjlraft 3.513217 2.317227 

Rule I. Rem. = 2.30^196 Rem. = 0.850891 



Chap. I. InSces Logarithms." io^ 

Ex. HI. From 3- 5 1321 7 Ex. IV. 0.850891 

, fub. 5.816413 3.168 1 18 

RuleL Rem. =,7.696804 Rem. = 7.682773 

Ex. V. From .7.503617 Ex. VI. .94.420345 
fub. .3.467306 ''9^^^73^57 

RijleILRem.=4.0263ii Rem. = f.746888 

Ex.VII. From .3.467306 Ex. VIII. .92.673457 
fub. .7.503617 .94.420345 

RuleII.Rcm.=.5.963689 Rem. = .98.2531 12 

. 

Ex. IX. From 5.81 641 3 Ex.X. 6.2067347 

fub. .8.132700 .94.1535012 

*i I I i 

RalellL Re.= 7.683713 Rcm.= 12.0532335 

Ex. XI. From .8.132700 Ex. XII* .94.278769 
fub. 5.8 1 641 3 6.165348 

RqIcIII.1 -~ RalcIII.l 

Part2. 5Re.= .2.3i6287 Part2. JR.=.88.II342I 



Multiplication ^Logarithms. 

6. In Chap. IV. Art. 19th and 20th of ihtTleory^ 
the Rules for multiplying the Indices of Logarithms 
(of Pure Fra£iions efpecially) are demonftratcd j and 
are, 

^ RuU I. If the Indeoe he Integral^ multiply as ufual ; 
the Produa (hall be Integral. 

RulelL If the Index be DeciihaU make the Lo- 
garithm oi Unity ^ ortA=:ioo, then (hall the Index 

P be 



io6 ' Multiplication of the 

be of the larger Sorf, wliich in this Cafe, will be more 
convenieftt forUfe ; all4 theti accbrdingas yoil muU 
tiply by 1, 3, 4, 5, 6, 6fr. you inuft rcjcft loo^ aoo, 
306, 400, 506, (^c. from the PrcduH^ the Remtun^ 
der thereof will be Decimal 

Ex. L Mult. 3.420673 Ex. II. 5.700672 

br 2 ^ 

Rulcl. P/od. = 6.841346 Produfts 22.8o2l68» 

■ ■ I 111 ■■•j 1 

Ex.III.Mult ,96.130126 Ex. IV. .91.034106 
by 2 i 

Rulell. Pr.= .92.260252 Prod.s: .73.102318 

Ex. V* Mult. .84.034121 Ex. VL .70.06105a 
by 4 • ^ £ 

Rule II. Pr.= .36.136484 Prod. = .SO.gc^S^S© 

In this laft Example, the Index .70. x 5=350 ; but 
iince from 350 you cannot rcjeA 40C>» as p^^KuIe II % 
therefore it muft be 400 — 350=. 50. which fub- 
dufted from 99, leaves 49 \ which ihews that 49 
Cyphers above 100, that is, 149 Cyphers are to be 
prefixed \ and this you a re to underfetnd in all Cafes 
where xL is lefs than af~ixtA. Sec Chap. IV. 
Art.20th,of TZ^^^ry. Or thus, in general lettA = 10 
or 1 00 ; then when ^L is greater than x — i x t A, then 
It will be ^L — X — 1 x tA — 9 or 99, is equal to the 
Number of Cyphers to be prefixM. But if ^.^i x tA 
is greater than xL, then it will be x-^i xtA — xh 
—9 or 99, =5= to the (aid Number of Cyphers. 



Df. 



i 



Chap. L Lidices iy Logarithms. 107 

Division ^I^oGARiTHMS. 

7, This is but the Rev^r/e of the foregoing Ope- 
ration ; and the Rule for decimal Indices the reverie 
to that 5 which is alfo derived from Chap. IV, Art. 
21ft, and 2 2d of the ^eory^ which iee. 

Rule I. If the Index be Integral^ divide as ufual ; 
and the ^otient-Index will be Integrals 

Rule II. If the Index be Decimal^ \xk iht larger 
Sort ; and then adding to the faid Index 100^ 200, 
300, 6?r. divide by 2, 3, 4, 6?^. the Quotient- A^ 
will h^ Decimal. 

Ex. I. Divide 6.841348 Ex. II. 22.802688 

by 2 4 

Rule I. Quot.=:3.42o673 Quotient = 5.700672 

^. m. PIv. .92.26Q252 Ex. IV, .83.102300 
, by 2 3 

RuleII.Qu.=.96.i30i26 Quot. =.94.367433 

Ex.V. Divide .36. 1 36484 Ex. yi. .30.305250 
by 4 B 

R»^c|I.QjJi=.84,034i2l quot. ::^ .86.061050 



^ 



p2 CHAP. 



jlo$ Mukiplicatm 

OOOOCSOOOOOCSO f^ OCSOOOOOOOOCQ 

C H A p. IL 

QTMuLTiPLicATiON ^»^ Division of 
Whole Numbers and Decimals^ I^o- 

OARITHMS. 

J. T TAving before fliewn the Method of finding 
JlX the Logarithms of all kinds of Numbers^ 
both Integers and Decimals^ and alio of fitting and 
adjufting proper Indices thereto ; and in the forego- 
ing Chapter, the Arithmetical Management thereof 
in ali Varieties : I fhall now apply the Ufe of thofe 
excellent Numbers in the Rules of Arithmetic ; and 
firft in the Multiplication and Divijion offFhole Num^ 
iersznd Decimals. 

2, From the foregoing Theory (fee Chap. I. Art. 
xo; and Chap. III. Art. 12.) we obtain this eafy 
and obvious Rule for the Multiplication of all kind 

, of Nijmhers by Logarithms^ 

f To the Logarithm of the Multiplicand^ 
viz. < Add the Logarithm of the Multiplier ; 
' "^ The Sum is the Logarithm of the ProduSI. 

Examples ^Integers. 

Logarithms. 

3. Example I. Multiply. ... 12 =s 1.079 181 

by 8 =! 0.903090 

Produift . • . ^ ^6 zn 1.982271 



Chap. It '^Logarithms. jtof 

Logarithms. 
Pxamplc n. Multiply ...... 127 aqc ^.103804 

by 12 =r 1.079181 

Pfodud. .... 1524 = 3.182985 

Example III. Multiply ..... 526 =ss 2.720986 
by 100 =3 2.booooQ . 

Produft .... 52600 r= 4.720986 

Example IV. Multiply .... 9876 t= 3.994531 
by , . . . 517 = 2.713496 

Produft . . 5105892 c= 6.708071 

Example V. Multiply . . . 9^7 600 » 5.994581 
by ..... . 517000 = 57^3590 

Produd: 510589200000 ==11.708071 

4. Examples o/Mix^d Numbers. 

Example VI. Multiply ...... 7,5 s=s 0.8750$! 

by 10 = 1,000000 

Produft 75 = 1.875061 

Example VII. Multiply .... 124 = 1.093422 
by, 3»6 ss 0.556302 

l^roduft . . . 44*64 =s 1.649724 

Ex- 



'lit MukipUcation 

^.ogarithmi. 
Example TIIL Multiply. . 0,762 » .9.881955 

by / 570 == 2755875 

Produa • . 434,34 c= 2.63783q 

Example IX. Multiply • . . • 36,5 == 1.562293 
by . • • . , 0,00019 =s .6.278754 

Produft .49^006935 = .7.841041 

"" ■' < 

Example X. Multiply 473 = .9.674861 

by • • h ^ 1.803705 

Prpdu<9: • . . » . 39.1 =: i.4785§6. 

Example XI. Multiply ..... j? == 0.823909 
by ^ ^S — •9744727 

Produdt 2'>y^ == 0.5686^:^6 

Example XII. Multiply . . . 2^,23' = 1.3 2 6541 
by 4^^ — 1.623458 

Produfl: ^00,71^ = 2.949999 



5. Examples ofVunt Decimals. 

Example XIII. Multiply .... ,12 = .9.079181 
by ...... . ,8 = .9.903090 

Produfl: .... jOgiS = .8.982271 

Ex- 



I 



Chap. II. h Logarithms. fit 

Logarithms, 
Example XlV. Multiply . . ,0097 ss .97.986772 
by ... • ^00021 ss .96.322219 

Produft 000002037 =s .94-308991 

Example XV. Multiply ,oo4^jz{ = .97.623458 
by . . . ,0000^ = •95-948847 

III II ■ 11 iw^ II ■ *m^mimm^mmm^mm,ami 

Produft ,0000003735 «3 -93.572305 

Example XVL Mult. ,00000085 i=: .93.929419 
by i66Gooi2 3= .94.079181 ' 

I ' ' I II . •- • mtkm ...■. III! ■ 

Produft ,00009000000102 := .88.008600 

Ex. XVII. Mult. ,00000000075 = .90,&75o6i 
by ,000006 = .94.778151 

H I I . ■ I M I 1— — »— .1.— i— ..-■■■—^—i^ili^M-^ 

Produft ,0000000000000045 as .85.653212 



6. DivisioK ly Logarithms. 

In the fame Part of the neory reftrr'd to (Art. 2.) 
for th^ Rule by which Multiplicatiim is performed by 
Logarithms^ you will likewifefind the Demonftmtion. 
of the following Rule of D/vi/fwi ofNumbcrt by 
Logarithms \ 

rFrom the Lorarithm of the Dividend^ 
. \subftraft theLogirithinofthePivifor; 
w2.<Yj^g Remainder is the Logarithm of th« 
L §u9ti<nt. 



Examples of Ik TJ^GZKs. 

Logarithm!. 
. y. Example I. Divide . . . /. g6 =s 1.982271 
by 12 = 1.079181 

Quotient . . . • 8 = 0.903090 



Example II. , Divide 1524 = 3.182985 

by . . • 172 a=5 2.103804 

Quotient 12 ess 1.079181 

Example IIL Divide 52600 ae 4.7209S6 

by 526 = 2.720986 

■II I I h— .111. ■ 

Quotient . . . • 100 = 2.000000 • 

Example IV. Divide, . . . 5105892 = 6:708071 
by 5^7 "^ 2.713490 

Quotient • • . 9876 = 3.994581 

Exam.V. Divide 510589200000 3= 11. 708071 
by . . 987600 =- 599458 1 

Quotient 517000 ss 571349^ 



8, Examples in Mix*d Numbers. 

Example VI. Divide ....... .75 =» 1.875661 

by 7»5 .?== Q.875061 

■ I ■ ■ wit 11 ■ 

Quotient • • . . • 10 ■= 1. 000000 
1 Ex- 



Chap. II. ^LobAftiTHM^; irj 

^ Logarithms. 
Example VII. Divide ..... 44>64 = 1.649724 
by ...... .. 12,4 = 1.09342a 

Quotient • ; . 3,6 = 0.556302 

Example VIII. Divide 434>34 = 2.637830 

by S7^ = ^'7^5^75. 

Quotient. . ,;^62 == .9.881955 

Example IX. Divide . . . ,006935 = .7.841046 
by 36,5 = 1.56229 J ^ 

Quotient ,00019 ^ -6.278753 

Example X, Divide ". . . 30,1 = 1.478566 

by ...-.,... . §^ = 1.803705 

tjuotient ..... *473 == .9.674861 

Example Xii Divide ....... 3^,7^2$ = 0.568636 

by ...;,.. ;_ ^ = 0.82390^ 

Quotient / =± -S-IAAT^J, 

Example Xil. Divide. . . ^60,71^ = 2-949999 
by ..... . 4i^P' ^ 1.623458 

tiuotient '. , %h'^% = \Z}^S^\. 



ti4 Dmfioa 

9* Examples in Pure Decimals. 

Logarithms. 
Example XIII. Divide. . . . ,096 = .8.982271 
by i ..... ♦ ,12 = .9.079181 

Quotient .... ,8 = .9.903090 

Exam XIV. Dmde ,000002037 =a: .94.308991. 
by ... . ,0097 =3 .97.986772 

Quotient ,00021 =: .96.322219 

Ex.XV. Divide . . ,0000003735 ~ .93.572305 
by 0000^ =s .95.948847 

^Quotient ^>oo4?f0 = -97.623458 

Ex,XVI. Div. ,00000000000102 =2: .88.008600 
by . . ,00000085 := .93.929419 

■ i III ■— I I I ' ■■ 

Quotient ,0000012 s= .94.079181 

Ex.XVII. DIv. ,0000600000000045 =.85.6532 1 2 
by.. ' »ooooo6 ^.94.778 151 

Quotient ,00000000075 =.90.875061 

io. I thiAk thefe Examples in the MuUiplication 
and Divifton oi Numbers hyLogarithmSy afe fufficient 
to m^xxxd: ?Ltij docible Genius in his Praftice herein ; 
and as the latter are but the Converfe of the former, 
fo they mutually illuftrate and prove the Truth of 
«ach ether refpeftively^ 

^ CHAP, 



viz.i 



ChzpJJLRatfingPowirityLoQAiBLiTnMS. 115 

chjlP. hi. 

OfraifingYovi'g'BiSy and the Extract lov tf 
Roots by Logarithms. 

I, "ITtROM theTheoiy of Lc^rithms (Chap. I. 
J|7 Art. 12. and Chap. III. Art. 13O wc have 
an evident Rule for the Involution of jffm^^Sj or 
raifing them to any propofed Power by means of Lo- 
garithms \ which is this, 

f Multiply the Logariibm of the given Number 
by the Index of the Power, viz. 2, 3, 4, $% 
(^c. the ProJuff {hall be the Logarithm pf 
the Power, viz. the Square^ Qube^ Biqua- 
drate^ Surfolid^ &c^ Power of the iaid giiien 
Number. 

Examples in iNyoiuTirfM. 
2. Example L What is the Square of the Num- 
ber 32? 
Multiply the Logarithm of . . . . • 32=i.505i5a 
by the Index of the Power . * . . . 2 

TheProd. isthe Log. of the Square 1024=3.010300 

Ex- II. Required the Square of 3,2=0.505150 

Multiply by • ^ 

The Prod, is the Log. of theSquare 10,24=1.010300 

En. III. Required the Square of . . ^^2:^.9.5051 50, 
Multiply by . . ^ » 

:{'hsPi;o<!luai».th^A;ifwer . . ,i024=.9.oio30o. 



Xi6 OfraifingFw^eri^ 

Example IV. Required the feveral Po^/frcrs of the 
Number 1.05 to the Surfolid? 

1. The Logarithm of 1.05=^.021189 

Multiply bjr .•..,.. . 2 

The Pxodua is the % Jr^ . . . 1,1025=0.042378 

2. Multiply the Logarithm of 1.05=0.021189 

by 3 

' ';|phe Produft is the Cube 1,157625=0.063562 

3. Multiply the Logarithnd of 1.05=0.021189 

by ...:.. 4 

Frodud: is the Biquadratt I.ai65o6z5=:0.o84756 

4. Laftly, Multiply the feme . . 1.05=10.021 189 
by ^s^ 

' VroduAistht Surfolid 1.2773315625=0.105945 

Example V. Reqi^ired tjhe Jfirfilid ]poiyer of the 
Number ,0006 } 

Multiply the Logarithm pf 30096=.96.77i8 151 
By the Index of the Power 5 

The SurfoL ,00000000000000007776=. 8 3,. S^qjss 

Example VL What is the CuboXuheVo^tx of ,08 ? 

Multiply the Logarithm of. . ,o8=.9 8.903090 
by the Index of the Power ... 6 

The Cuh'CuieFov9tt ,000000262 144zz.93.41 8540 

Example VII. What i§thc 57th Powenofthe Num- 
ber 399 ? 

Multiply 



j 



Cliap. III. 6y lyOGARiTHMs. 1 17 

Multiply the Logarithm of . . . 599=^.9.995635 
%y th? Index of the J^ower ... gj 

69969445 
49978175 

The 57th Power IS . • . ,56389, &c.=.9.75i 195 

3, There is another way of raiBng the Powers of 
Decimal Numbers by Logarithms, and it is thus 5 

r Multiply the Aritbtnetical Complement of the 
1 Logarithm of the given Fraftion by the 
viz. < Index of th^ Power, the Arithmetical Com- 
I plement of the Produdi: is the Logarithm of 
t. the Power fought. • 

Apd this in many Cafes, (as when the Xndex pf the 
Power is a mix*d Number or pure Decimal) will be 
fpund moft certain and ready. Thus in the laft £xt 
ample this way ; 

Exarpple VIIL What is the 57th Power of the 
Number ,99 ? 

The Logarithm of 399=-?-995635 

The Arithmetical Complement 0.064365 

whiqh multiply by . . ^ gy 

■ . ^ 

21825 



The Produfl: . . . 0.248805 

The Arithmet. Comp, is 1 ^r^o^ «,^ ^^ ^ 
the Log. of the Power J '56389> ^^.=^9^751^5: 

' ^xamplelX. What i« th^lth or 25th Power of ,z ? 

Tha 



JiJ J^:ictra£fion of RB0ts 

The Logarithm of . . . ^aszi.p.goiojo 

The Arithmetical Ccmp. thereof ,0.698970 

which multiply by the Index ,25 

3494850 
1397940 



TheProduft . . . 0,17474250 

The Afithmet. Comp. is j ,^0 o 
the Log. of the Power \ '^^^74, 8{c. =.9.825258 

Example X. What is the 6,25th Ppwer of ,0032 ? 

The Logarithm of . . . >oo32=:. 7.505 150 

The Arithmetical Complement 2.494850 

which multiply by . . .' ^ 6.25 
• 

The Produft is . . . 15.5928125 

The Arithmetical Complcm. of which is .84.4071875 

And the Number anfwering thereto, viz. 
3O0000000000000025538 is the 6.25th Power of 
5O032. 

4* EvotuTioN^r Extraction <?f Roots 
^j' Logarithms. 

This is done by a Rule, the converfe of that fovlnvo^ 
lution^ in Art. ift 5 

{JDivide the Logarithm of the Power by the 
Index of the Root, the Quotient (hall bQ 
the Logarithm of the Rootlbught. 

Examples inRvohvr 10 u. 

Example I. What is the Square Root of 1024 ? 
Divide the Logarithm of . , . 1 024=3. 01030Q. 
By the Index of the Root 2 

The Qudt.is the Log .of the fquare Root32= 1.505 1 50 



Chap. III. by Logarithms. i 19 

Example 11. Required the Cube R$ot of 1,157626 ? 
Divide the Logarithm of 1,157625=0.063567 
By the Index of the Root 3 

The Qu. is the Log. of theCtf^^ R. 1,05=0.021 189 

Example III. What Is the furfoUd Root of the Power 
,00000000000000007776 ? 

The Logarithm, thereof is . . . *^i'^9oy5S 
which divide by Ae Index 5 

Example IV. What is the" Guho-Cuhe Root of the 
Power ,000000262 144 ? 

The Log. of . . ,000000262 I44c=.93.4i8540 
"Which divide by the Index 6 

The Root fought is . . . ,o8«=.98.903090 

Example V. What is the 57th Root of the Power 

The Logarithm of . . . ,56389> &c. .9.751195 
which divide by . . . 57 

The Root required is . . . >99='9*995^35 

5. Another different way to extt»ft the Root of 
Decimal Numbers is the convcrfe of that in Art. 3d, 
hereof. 

r Divide the Arithmetical Complement of the Lo- 
I ;garithm of the Decimal given by the Index 
viz. \ of the Root required, the Arithmetical Com- 
i plemcntofthe Quotieut is the Logarithm 
L of the Root fought. • 

Ex- 



1.2b EkfjraSliondfRobtiy &c.' 

Example VI. What is the ,25th Root of the Po wet 
,66874, &c. ? 
The Logarithm thereof is . . . .9.825258 

The Arithmetical Complement is 0.174^42 

which divide by . . . ^25 

The Quotient . . , 0.698970 

The Arith. Comp. is the Log. 7 ^ ^^ ^^ , ^ .^ 

of the Root "^ ^}- • •2=-9-30i030 

Example VIL Required the 6.25th Root of the 
Power ,00000000000000025538 ? 
The Logarithm thereof is . . . .84.4671 8 Jr 

The Arith. Comp. of which is 1 5.5928 1 3 

which divide by . ; • 6.25 

The Quotient is . . . 2.494850 

the Arith. Complem. thereof is . . . .7.505i50 
the Logarithm of ,0032 the Root fought. 

Example VIIL What is the CuboXube or 6th Jloot 
ofthcPower ,1 ? 
The Logarithm of . . • ,i=.9.ooooo6 

the Arith. Comp. thereof . . •^•999999 

which divide by the Index 6 . 

The Quotient ... 0.166666 

The Arith. Comp. is the 1 ^o,^^ o, o 

Log. of the R^ot i .68129, &(;, =.9,833333 

6. Thus you fee the great Ufe of Logarithms iri 
cxtradliog the Roots of a given Power, which tho* a 
thing fo very difficult by the Rules of common Arith- 
metic, is yet render*d moft eafily prafticable by this 
excellent Art ; yea 'tis eafy to make it appear, that 
the ExtraSion of Roots is not only moft expeditioufly 
performed, but hath a more univcrfal Perfeftion in 
this Method, than in any other. 

chap; 



chap* IV. Faridusttukstf Proportion, ?21 

CHAP.' IV. 

Of the various Rulis g/'lPROPORtioN, and of 
finding Mean Proportionals ^y Loga- 

RITHMSk 

* • 1? R O ^ *^* ^wry *t>* evident, that the golden 

J[7 ^«^^ Of ^'' of Proportionalsy is wrought 

in Logarithms by only the Addition and Suhftra&ien 

of the logarithms of the Terms of the Proportion. 

And {f the Proportion be direA^ the Rule is thus ) 

r Add the Lx)garithms rfthe fecond and third 

. . \ Terms, from that Sm fubftrad the Loga- 

^'^•] rithm oiihajirfti the Remainder is the 

£ Logarithm of the fourth fought. 

Examples in the GoldAn Rule Diteff, 

Example Iv Ifit2Pbundscoft il.iss.$d, what will 
173 Pounds coft? . ' 
The Logarithm of . /. I2=:i.679i8i 

f o the Log. of i/. t^s. 9 i.=i,7875=so.252246 
Add theLo^thm of 173 =2.238 046 

theSum..... 2.490292 

Subfttaa the &ft, there refnains 2^,77 /. =i.4inii 

Wherefore the Anfweris 25,77 /.=25/. *5^- 4<^.i» 

2. But fince if ,you divide by any Number, or 
multiply by its Reciprocal, the Eflfeft is the feme j 
andalfo fince the Arithmetical Complement of any 
Number, is but theL©garithm of the Reciprocal oi 



122- Of the varioui kuks 

that Number; therefore it follows, that where the 
Sf/^ra&i(m of a Logarithm is required in any Ope- 
ration, if you take die Arhbmettcd Cemflement eS 
that Logarithm, the^hole may be performed by 
Addition only. Thos m the foregoing Example. 

{the Ar. Comp. of the Log, of 1 2.=. 8.9208 19 
the Log. of •^. 1,7875/.= 0.252246 • 
the Log- of ^ 1732x2.238046 

The Sum is the Anfwer • • • . 25,77 '•=141 ^ m 
the fame as before* 

Example II. If 2C. iq. 21U i^4fz.CQ&5t. ijs. tiX 

what will 31 C. zq. 26 1. 150%. coft ? 
Then 2 C. iq. mL i4^.:s:a.4453A.Coiit. .9.61 1668 
The Log. of $1. lys. Sd,i^ 5.8844 = 0.769702 

« 111 II 
The 5«iw IS the Logarithm of 76,38 i/.=:i. 882984 
Therefore j^^iiL^xfuSL js. yd.i is the Anfwer. 

3. . 0//&r Role of Three /»wf/Jr. 

In this Caie you mufl take the Arithmetital Conh 
flement of the tbifd TerW, and add it with the Lo* 
gu-ithms of the other two asJbefore \ ib ihall the 
Sum be the Log^ithm of the Anfwer. 

Example* Suppofe a Field feeds 18 Horfes for 7 v 
Weeks, how long will it feled 42, at that rate > 

{the Logarithm of 18=1.255272 
the Lo^rithm^of « 7=0.845098- 
the Arim.Comp. of the Log. of 42=.8.37675 1 

The Sum is the Lc^arithmof Anfu^r 330.477 12 1 

4.: 



f 



Chap. IV, ^ Prfifartim. » J23 

4. Oftbi dotthle Rule f^tbru^ cr Rule pf 
Five Numbers. 

As in QjiefKohs of this iQrt» diete axe always tbue 
idndiiwHaToc Jufpofed Terms ; the £rft of which is the 
principal Caufe of Gain^ Lofs^ ABitm^ &c the fecmd 
^ denotes the 7/i»r, Difiance^ &rc. and the third is the 
Gain^ Lofsy or AEtitm^ &c. So kt thefe three Terms 
be denoted bjr the C34>ital8 P, T, G. Alfo there are 
three other Terms (fimilar to the three former) which 
make the Qjieftion to be lefolv'd % and let thefe be 
wprefented by the fmajl Letters pt t, g. Two of which 
are always given, and the other is fought. But fince 

P.-Girpi-^^ and again, fincc T:^::t:gi 

therefore Tg = ^ , and confcquently PTg=tpG ; 
from which general Theorem we am eaftljr find p» t; 
or g. Thus, 1 i^ = Pi and W.^^izxAHl. 

J~ as g. The Contrivance of thefe excellent The- 
orems weowe to the fete Mr. Wari^ cXCheJtir. 
5. I fliall exemplify Queftimski this Rule by Ex- 

.amples, as follow. 

Example I. If 100 /. ini 2 Months gain 6 /. what will 
350/. gain in 9 Months ? 

Here P=ioo, T«:i2, Gx=s6 j alfo p?=350f t=s9, 
to find g. 

f the Logarithm of, ..•.., G=;=6=:0.778i5i 
Add< the Logarithm of p=3 50=2.544068 

(^the Logarithm of t==9=?o.954242 

The Sum is the Inqgantbm of « • . ^Gpfc=4-2 7^461 

R 2 Add 



i 24 Of finding Mean Proportionals 
^•,c the Logarithm of . . • . . P==ipoi=:2.oooQOp 
I the Logarithm of T=i2=i.©79i8i 

The Sum is the Logarithm of pT=;3.o79i8i 

Then from the Logarithm of ... • Gpt=4.27&46i 
Subdudl the LogOTthm of PT=3.o79i»i 

There rem. the Log, of ^5=g== 15,75 =1.197280 
Wherefore the Apfwer is i^.^sl^i^l 15/. 

6. Examxple IL If .36 Bufheb wijl ferve 24 Horfes 
48 Days, how long will 126 Buftiels ferye 
96 Horfes ? 

Here P=s24, T=48, Gc»3(>; alfQp=96, and 
• g=i26i to find t. - 

f the Logarithm of P=24=:i .3 802 1 1 

Add^ the Logarithm of T=4fc=i.68i24i 

l^the Logarithm of g==i26=?. 100370 

The Sum is the Log, of . - PTg=r5-j6i822 frpm'j, 

i. J. C the Logarithm of G==36=|. 556302 '. 

■ c the Logarithmi of pc=96=i. 982271 | 

The Sumi is the Log, of ... . Gp=:3. 538573 fubf.^ 

Thefefo|« Ae Lag. of H^ = t = 42 = 1.623249 
ThcAnfwcr, viz. 42Wec^s. 

7. Example III. At the I^tc of 4s/. per Cent, per . 
Am. what Principal will produce' 35/. 15^. 
in 7,5 Months ? 

Here |*=|oo, Tsb:i2, Gssi4,5 j alfo tssr7.5, and 
6=25-7^ « to find p. 



J 



Chap* IV. by Logarithms. %z$ 

{(be Logarithm of P=:ioQ=2,ooooop 
the Logarithm of T=i %^\ .079 1 8 1 
the Logarithm of g=35,75=i.553^76 

ThcSum is the Log. of . . . PTg=4.632457 fromy 

- , J 5 *^* Logarithm of 0=4.5=0.653^2 I 

^^ \ the Logarithm of 1=7.5=0.8 75061 j 

The Sum is the Log. of 01=1.528273 fubcj; 

Whence the Log. of^?=p=i27i ,1 I4=3.i04i84« 

Confequentiy, I27i,ii4/.=i27x/. 2j. 3^4 is the 
Anfwer. 

NotCy Thcfe Theorems give the Anfwer ahTplutetyr 
without regarding whether the Proportion be DireS 
or InvirfCj or both together, as in Art. 6. Exam. 1I« 

8. Of the Method of finding Mean Proportionals, 

In* order that a clear Notion of this moft uftful 
problem may be had, I (hall premiie the followbg 
things. 

1. BtVHtexi two fquare Numbers AK and BB, there 
will fall but one Mean Proportional AB ; that is^ 
A*: ABS:: AB:B*. SetEucl.i. ii.: 

2. Between two Cubic Numbers A^ and B*, there 
will fall two Mean Proportionals A*B and AB* ; that 
is. Ah A^3;: AB* : B». See Eucl. 8. 12. 

3. Between two Biquadrate Numbers A^ and B\ 
there will fall three Mean Proportionals A'B, A*B\ 
and AB» •, that is, A-^ : A«B :: A^B : A*B* :: A*B* : 
AB» :: AB^ : Bl 

4. Again, between two furfolid Numbers A^ and 
B», there will fall four Means^ viz. A^^B, A^B% 
A*B% and AB^j that is. A*, A^B, A'B\ A*BS 
AB% BS will be Proportionals. And thus the 

I Number 



ri26 Of finding Mean Freportionah 

Number of Mean Proportionals will tc always le/s hf 
one than the Index of the Fowcr of the 0ven Ex- 
tremes. 

9* But the comrnm Ratio of all fuch Seriea 13 \ 



ForAAxI =AB; tndABx \ «BR 



AndA«x?.=A*Bian(JA*Bx-? =AB*; and 



AB* X 1- sc:B*. and fo in the others. Here I have 
A 

fuppofcd A tobe the Jcaft Number, and B the greateft, 

and the Scries to begin firom A*, A*, ^e. But if B 

be lefs than A, and the Series begin from B*, BS fcf <•. 

then the Raeio of the Scries will be •g • 

Now| isRootof ^,,5^, 5>^^'*^^^- 
fbre from thefe Prcmifes well underftood, 'tis ca{y to 
(:onceive the Reafon of the following Rule for fold- 
ing Mean Proportional?, vi%. 

"SubftraA the Logarithm of rhcfe^j^firwfrom 
the Logarithm of the greatefiy and divide 
the Remainder by a l^urnb^ greater by one 
than the Number oi Means defircd ) then 
Ilule^ add the Quotient to the Logarithm of the 
Icaft Term {or fubftraft it frofh the Loga- 
rithm of the greateft) continually, and it 
will give the Logarithm^ of ^ tl^e ^Seavi 
Prop4iriumals defired. 

10. Example L Let two Mean Proportbnals b^ 

fought between 8 (=A^) and 28 (=;; B\) 

♦i' ■ 
The 



Chap. IV. bf Logarithms. iny^ 

The Logarithm of B*sss28aBX.44f7i5S 

The Logarithm of • • ^ . • . * A^:te8=^o.903O9o 

The Difference Is ^, so.544068 

Which divide by 3 

B 
The Quotient is . . • "J =0.181356 

To which add the Log. of . , . A^s8sBO.9a309o 

■^.ft^ th. Log. of J A'B-»,.^l.o«444« 

To whkhaddagain -^ sso.iSijfS. 

\ 

The Log. of the 2d Mean AB*si8»44=si.265loz 

■ -,i 

Wherefore thePropor- > 8 : 12,14 :: 18.44 • 28 
tionakaie S A* : A*B :: AB* : B*. 

XI. Example II. Between x 6 and 64 find £ve 
Mean Proportional^. 

The Logarithm oif B*=r54»i.8o6i$c^ 

Subdudthe Logarithm of. . A^ssx 6s 1.204 120. 

B^ 
There remains ^ . . ^ =:p.6o2o6t> 

B 
J of whichis -^^ =0.100343 

To which add the Log. of A*s;:i6=;i.t04i2o 

*l'T.S^^- "'l A.B=.o..5b...3a^3 
To which add again ^ :so.ioo343 

The Log. of the ad Mean A*B*ia:25.398tti.4048pfr 

And 



J28 Simpte Mereji 

And thus you produce the Logarithms of^ 

{3d Mean ; . A*B»=32=i.505r5<j 
4th Mean A*B*=40.3i7=:i.6o5493 
5thMcan *. . • V . . . AB«=50,796=i.^05836 



-«Ei- 



\ 



The Series tlierefbre is this, 
15.20,158/25,398. 32v 40,317- 50»796- i4- 
A^ A>B. A4B\ A^B\ A^B*. A'B. %\ 



t2. If it were required to find 3^4 Mtan Propor- 
tionals between o and i .06 ; or o and i .05 ; or o and 
1,04, &fr. then A***=o, and fi^**=i,o6i 1.05, 

Cfr. and fo -^ ^^^s/ZoSl otVlioJ, tff* 

Wherefore if you multiply the Logarithm of B, by 
2> 3» 4» 5» &c. to 365, you will thereby obtain 
the Mean Proportionals rtquittd. And thefe will be 
the fcveral ^mounts of i /. and its Intereft^ for each 
Dtfy of the Year, at the Rates of 6, 5, 4/. &c. per 
Cent, per Annum^ Compound Intereji. But more will 
be (aid of this hereafter. 



# 



C H A P. V. 

Simple Interest iy LogaJhtHMs* 

I. Ti ^ Y Defign being only to acquaint the Reader 
iVJL ^^^h the Theory and pra£lical Vfes of Lo- 
garithms^ and not to treat of the Theory of any other 
Art or Branch of Mathematical Science ; it will be 
fufficient for me barely to mention the Theorems or 
Rules^ on which the divers Parts of Learning (I 
(hall treat of; depend, and fhew how they are moft 
conveniently wrote by Logarithms. 

Of 



0/S I MP LE IkTZRSST. 

2. 1 have more than once ferv'd myfclf with thofe 
excellent Theorems of Intereft contrived by the late 
ingenious Mr. fFardi and ihall once again makis 
them fubfervient to my Deiign in this Place. In 
order to which 

f P=any Principal or Sum put to Intereft. 
J. \ R=the Ratio of the Rate^ per Cent, per Ann. 
1 T=the time the Principal continues at Intereft. 
I A=the Amount of the Principal and Intereft. 

3. Then any three of thefe being given, the other 
may be found by the following ^theorems. 

Thcpr. I. TRP+P=A. Theor. II. .j^ = P. 
Theor. HI. —^ = R. Theor. IV. ^ = T. 

Queft. I. What will zysL igs. amount to in 3^ 
Years, at 4iL per Cent, per Annum ? 
Here V=275yy5, T=3,5, R=o.045, to find A. 
Theor. I. 

{the Logarithm of . . . P=2 7 5. 755=2 .4405 15 
the Logarithm of • • . T=s 3.5=0.544068 
the Logarithm of. .. R=so.045=.8.6532i2 

The Sum is the Log. of PTR=43.43i=i-637;^95 
To which add P=275,75 

Queft. II. What Principal or Sum being put to Ip- 
tereft, will amount to, or raife 5 Stock of 
- 319 ^- 2^* 7^* i" 3^ Years, at thQ rate of 4^'* 
per Cent, per Annum ? 

^ S Th«. 



'f^6 Simpklnterefi 

The Log. of the Amoum A=/. 3 1 9, 1 8 i=a .50403 6 

- jjCthe Log. of the Time . . . T=3,5=o. 544068 
• 1 the Log. of the Rate R=o.045=. 8 .65 3212^ 

The Sum is the Log. of . . TR=o.i575=.9.i9728q 

Then the Log. of ... TR+i=;=x. 1575=0.063521 
Subftraded from the Log. 1 ti ^^ ; * 

of the Amount leaves ^ I P=275>75=2.4405i5 

yrhcrefore the JV7»f/>j|} ,„^- ^-_«^^; ,^, 
required is \ Uy5'75=^75l' ^5s. 

Queft.III. At what Rate ^frC(f»/.&c. will 275/. 15 J, 
amoant to 319/. 35; yd. iii 31 Years ? 

Here A-:-P=3 1 9, 1 8 1—2 75,75=43.43 1 . 

The Logarithm .... A — P=43.43 1=1. 637800. 

...^t the Logarithm of . . . P=i2 75,75=2.4405'! 5 
- I the Logarithip of . . . T= 3,5=0.544068 

The Sum is the Logarithm of PT . , . =2.984583 

* Queft. IV. In what Time will 275/. 15 jr. raife n. 
Stock oif 319/. 3j. 7<;. at the Rate of 4I/. 
fer Cent, per Jnnum? 

■^ \ the Log. of the Ratip R=0,O45=. 8.6532 ? 2 

The Sum IS the Log. of PR= 1.093727 

Which fubftraft from \ a p_, ^ , ^' _, a;.^q^^ 

the Log. of I A~P=4M3i=i.6378oq 

Thene^ 



chap. V. by LooAliitHMs. 231; 

There will remwi the j A-^p 
Log. of \ "^=^=3^5=0.544071 

Therefiwc the Anfwer is 3* Ycais. 

^ 4. Of Annuities, Csfr. in Arrears. 

!rtJ=the Annuity^ Penjion^ or yearly Rent. 
T=the TVfw^ of its Continuance iinpsiid. 
^ut / R=the Raiio of the i?/?/^ of Intereji. 
. I A=:the Amount of the Annuity and its tntereft; 

Then the theorem f<ir finding each of thofe Parti* 
tulars, are as follows. 

Theor.t. I2izSJxR:+TU=A. 

Theor.II. xrk— tr+zT =^* 
Theor. III. yr(;f_yt) = R. 
Theor. IV. ^^W^ • — i'' =*^T. 

Queftioill. If 250/. ^^ar/y Rent (Penfion, €9*f.) be 
fbrbom or unpaid 7 Years, what will it amount 
to in that time at 61. per Cent, for each Payment 
as it becomes due ? 
Her6Uss250, Rss=o.o6, Ts» 7 j to find A^fer 
Theor. I. 

- , J 5 the Logarithm of .... 17=250=2.397946 
'^'^'^i the Logarithm of . . ; . Ta«= 7=0.845098 

The 5«<» is the Log. of .. TtJ=:i750ta3.24303d 
add the Logarithm of . . . T=« 7mo.845098 

The Sum is the Log. of TTU=i225oai=4.o88i28 

S a Then 



IJ2 Simple Intireji 

Then the Log. of '^"— "^ =5250=3.720159 
to which add the Log. of Rs=o.o6i=:.8.778 15 1 

The Sum is the Log. of^^SzSJxR— 3 15=2.4983 10 



To which add TUs:i75o 

TheSumisHHrlH? ' ~ , , 

„ ,^,\ t 2065/. the/ 

xR:+TU=A= 3 

Note, if thefe Payments be made 
rQjMjrterly. -)^ pR, iU, and 4T. 



' Half-yearly, J. « <4R, 4U, and 2T. 
(Thrce^uarterly,J^ (|R, iU, and IT. 

Qucft. II. What Tear-Renty Penfion, &c. being for- 
bom or unpaid feven Years, will raife a Stock 
of 2065/. allowing 6 per Cent per Annum, 
for eadi Payment as it becomes due ? 

Here A=2o65, TaB7, Rs=o.o6 j to find U, per 

Theor.If. 
f^^^\t\itIjo^ihmo( Tss/sb: 0.845098 

I the Logarithm of . . . RssO,o6ss:.8.778i5i 

The Sum is the Log. of. . . TRs=:o,42=.9.623249 
to which add the Log. of T=:7-5 0.845098 

The Sum is the Log. of . . TTRsa2,94= 0.468337 
From which fubdu^l: . . . TR=o,42 

There remdns . , TTR— TR=2.52 
to which add ...... 2Ts=:i4 

the Sum is TTR— TR4.2T=:i 6,523= 1.21 8010 
the Lo^rithm of 2 A=4i 30^=3.615950 

The 



Chap. V* ^Logarithms. 135 

The DiiFcrcnce of thcfc iHOgarithms is U=a25o/. sat 
2.397940, which is the Annuity fought 

Queft. III. If 250/. Yearly Rent, 6?f. being forbora 

7 years, will alhount to 206 5 1. allowing 

iiiAple Intereft for each Payment as it becomes 

'^^ due, what muft the Rate of Intereft be pir 

Cent, per Annum ? 
-. , . C the Logarithm pf . . . . U»250aB2.39794a 
t the Logarithm of . • • . Ts3s7s=s 0.845098 

The Sum is the Log. of . . . UTr-i 750=8:3.243030 
to which add the Lc^. of . . . Ta7ss 0.845098 

ThfSum is the Log. of •TTUr=:i2250ar4.o88i28 
from which fubduft .... TUs=i750 

there remains TTU— TU=: 1 05005=54,02 1189 
the Logarithm of . a A — 2TU=:630s=52.79934o 

Subduftthe former from the 1 ^^ ^ ^^ o ^^o ^ 
kttcr,th«€ will remain } ^^0.06^.9. jjZi^t 

Wherefore, as x/. : 0.06/. :: 100/. : 6/. the Rate re- 
quired. 

Queft. IV. In what time will 250/. Yearly- Rent, 
raiie a Stock of 2065/. allowing 6 /.p^r Cent^ 
&c. for the Forbearance ef each Riymentas 
it becomes due ? 

Here 17=250, A=2o65, R=sio.o6, ^ — 1= ^ 
— i=3^>2^s=^5 to find T, ;>^r Theor. I V. 

- , J c the Logarithm of ... . U=250c=:2. 397940 
cthc Logarithm of . . . • R~p.o6=.8.778i5x 

m ■ n * 

The 



134 Simple Intereji 

The Sum is the Logarithm of RUi=:i 5^=1 . i y6og t 
ivhich fubf. from the Log. of 2A=:4i30=s:3.6i595o 

2A 
there remains ^ «275>3^=^-439859 

Addi *^ Logarithm of xzsaa,^'^ 1.509658 

(the Logarithm of . . * . i;c=8.o83'=d.907594 



XX 



The Sum is the Log. of 7- =261,3605=2.417252 
to which add ,..•.. Iir=275»3333 

the Sum is . . ^ + tT . =536,6938=32.729724 

Half that Dm. 2 ^^A - xx ^,^ ^ ^^ 

istheLo^ J RU + 1 =23,166^=1.364862 

from Which dedu6t . . * • ix=i6,i66^ 

there wiU temain "^^ + ^ : ~4x;=t=7, the 
Number of Years required. 

5. Of the Present Worth ^Ani^uities, 
Pensions, Gf^. 

Here U, P, R, T, are ufed to denote the AnnuHji 
prefentWorihi Rati$ of the Rate oilnterefi^ and 

^ime^ as in the former Articles ; here alfo let ■^— 

U — i=Af ; then the Theorems for Operation are 
as follows \ 

Theor. I. "^^l^^fc^I 5,u=P= Prfefent Worth. 



2TR+2 

ppf X 2P— U,=rAnnuit 

Theof. 



TR+t 
Theor. 11. ttR— TR+zT ^ 2P— U,=rAnnuity, 6fr. 



Chap. V. b} Logarithms. 135 

iP— •2TU 

TheorJII. TTiTZftjirSPT =R=,Ra^o of the Rate, 

Thcor. iv/^ + "J =±= ^^=T, = the Time. 

Qucft. L What is the prefent Worth of 75 /. Yearly 
Rent, to continue* 9 Years, at 6 fer Cent. 
Sec. 

Here U=75, T=9, R=o.o6, to find P, per 
Theor, I. 

^ ,j cthe Logarithm of T=9=o.954242 

'^^"1 the Li^arithm of R=o.o6=.8.778i5i 

The Sum is the Log. of . . . . TR=,54=.9.732395 

Alfothe Logarithm of. . TTR=r4,86=o,686635 
fubftraft TR=,54 

there remains . . . , TTR~TR=4.3^ 
tp which add 2T=i8. 

The Sum is TTR—TR+2T=2^,32=:j. 348694 
the Logarithm of . . . 2TR4.2=:3.o8=o.48855i 

the Difference is * =0.860143 

To which add the Log. of . . . 11=75=1.875061 
The 3um is the Logarithm of 

" ^aTlIz"^'^ X U = P = 543,506 =2.735204 
Wherefore the prefent Worth is 543/.. tos. id A. 

Queft.II. What Artnuity, Penfion, 6fr. may bei 
purchafed for 543/. 10 s. i^. I, to continue 
9 Years, allowing to the Purchafer6p^rO»/. 
ftr An. fimple Intereft ? 

Here P=543,5o6 5 T;=:9; R=o,c65 to find U, per 
Thcor. \h 

From 



136 simple Jntereji 

From th«5 "Log. of TR4-i=i,54=o.i8752i 

fuM^Acj TTR_TR+,r=..,3«,.348694 

to the DilRrence .8.838827 

add the Logarithm of 2P=io87,or2=3.0362j4 

The Sum is the Logarithm of 

fTll^yft+at X 2P = U =5 75 =1.875061 
That is, 75/. perjnnun^ is the Amiuity, fcfr. fought. 

Qucfl;.III. If 543/. 10 s. I J, J- ready Money, will 
purchafe an Annuity, Leafe, 6ff. of 75/. 
per AnnufHy to continue 9 Years ; ^uare the 
Rate of Intcrcft per Cent. &c. ? 

Here P=543,5o6; U=75 \ T=:o 5 to find R, jp^r 
Theor.UI. ' 

From the Log, of 2? — 2TU=:262,98 8=2.41 9936 
*L?g. rf} TTU-TU-2PT=4383,io8=3.64i782 

the piiftrence is the Lpg. of R=o.o6=.8.778 154 

Wherefore, as i/^: 0,06/. :: lOp/. : 6/. the Rate. 
rc<juired. 

Queft. IV. If for 543/. los. %d.il purchafe an An- 
nuity, Penfion, 6fr. of 75/. per jiHnum% 
^are^ how long I may enjoy it, at the Rate 
of 6 per Cent. &c. Intereft r 

JJcre P=543,5P6, U=75, R=o,Q$i tofindT,^fr 
Theor.IV. 

From the Log. of . . . !;}P=io87,oi 2=3.036234 
fubduftthe Log. of 17=75=1.875061 

2? 
the pifF. is the Log, of. . g- =14,493=1.161173 

WWch 



chap. V. by Logarithms. i^f 

which fubftraft from • . ^j -^;~32^33J 

there remains g- — j^ — is=s;fesi7,840=i.25i395 
add the Log. of 4^=446=0.649335 

the Sum is the Log. of, . 4xAr=79,5664= 1.900730 
to which add ^ =241,558 

the Sum is ^ + *^ =3321,1244=2.506659 

half which is J ^/Ip TTT 
theLog.of} ^Ku +-; =17,919=1.253329 

fubftrad ix=8,92 

there remains v^J^ 4.^''— ix=Ts=8,999=c9, 
the Years fought. 

6. If the Queftionbe of Annuities, &?r. in Reverfiony 
you muft find the Amount of the Purcbafe" Money to 
the time of G>mmencement (together with itslntereft) 
by Queft. I. Art. 3. and make that Amount the Sum 
for the Purchaie ; and then proceed as in the Que* 
ftions of this laft Article. Thefe are all the fund4h 
mental or original Czies of ^mpie Intereji. 



^ 



CHAP. 



CHAP. VI. 

CoMpauND Interest by Logarithms. 

I . ^T^ H E former Qycftions o^Jimpk Inter e^ ipight 
X be refolved by x\i€ ^\A€% oi vulgar Ariih* 
metic \ and I have only there (hewn, they may alfo 
be (and that in mwy.Ga^s, moft'convifiienfly) wrought 
by Logarithms. But fa the- prefent Affair of compound 
Jntcr^j the Ufe of Logarithms is ahfolufely necejfary ; 
no other Method pf the Solution of Queftions in 
compound Intereft being equal- to- it in- Extent and 
PerfcAion. And cbnfc^jcutly the young Student in 
Arithmetic is under an indiipeniible Qbli|ptioii.lx> be 
acquainted with this moft excellent and uleful Branch 
ot the Science. 

2 . I fcall here alfo- proccicd according tq the Theo- 
rems of Mr. fFard ; and therefore 

r Prrthe Principal put to Inter efi, . , 
\ t=thc ^ime of its Continuance. 
Put^ A=the Amount of the Principal and Inter0. 
• I R=:the Amount o{ il.znd its Interefi for one 
t Year, at any given liate. 

Note^ you find \ lOO : io6 :: i : i,o6=:R, at SperCt. 
^ R thus, J loo : 105 :: i : i,05=R, at ^per Ct. 

The Amounts of i/. c i. 2. 3. 4. 5, fcfr. Years, 
in fcveral Years \ R. R* R» R^ R^fcfr. Am.of i/. 

Therefore Rt = the Amount of i /. in the Time t for 
the Rate agreed on ; this being premifed, the Theo- 
rems for Compound Intereft are as follow. 

3. Thcor. I. Y'^'znh.^xh.t Amount, 

Theor^ 



Chap. VI. ^/LoGARtTHM^'; ^3^ 

Theor.il. I =?= the ?rinclt)ai. ■ ■ 

Thcor. III. ^ = R* I — the Amouotof t/. 

By thefe Theorems the feveral Cb^ftions «f Com- 
pound IntenJftrare ahfw^ed sftOft expeditiouay by 
the Li^aritbms in th« M»nnw folloMving. . 

4. Qgeft. I. What ^1 27s/. 15^- amount toin 
3! Years, at 4 1, fier Cent, per Jnnmi Com- 
pound Intereft ? " • 
HerePs!«275,75 4 Ra=i,04^i t:s=3,5'j ihtn^ A, 
ferTbeac,: I. 

The Logarithm of R»i,Q45:=:o.oi9ii6 

. multiply by the Time ... .t ^ . ^5 

■ The ?^. 19 the Log. of R'^i ,1665=0.066906 
To which add the Ldg. Of P— 275»75^^-4405i5 

TKeSumistheLog. of PR«=.^i2 1;,68==~2. 507421 
.. $0 the Amoppt fought is 22 i.Li 3^s. 7d.\ which i» 
m6te tJ^n thi'Amojvitlby finipre rntereffi by ij.jos. 
S*(^cff,I. Alt. 3. of ^he foregoing Clis'tJta^. ' 

... ■ . . -^ . : -■■J , -T 

Oueft. IT. WHat Principd or Sum ^Ing piV te Vfe 
at 41 p& C^. eoiiipdutid jWereft, will ■ 
ambfint to 32 1 /. 1 3 i. 7 d. in 3I- Years ? 
Here A=c32 1,68-, R=i,045i t=3'5 J to find P, 
per Theor. II. 
Frdm-ffieLogarithmof A=32i,68=2.50742i 
Subduft thel5>garithra of R'=i.i665=ao^69o6 

TheDiff.isthfiLog. of^.*=:P=a75»75?=f2-4405».5 
therefore the Principal fought is 275/. 15 J. 

. T 2 Q««ft- 



/ 



1 4» Compwnd Inter efi 

Qucft. III. At what rate pr Cent. &c. Compoun4 
Intereft, will 275/. 15J. raifc aStock, or a- 
mount to 321 /. 13 J. 7^. in 3I Years ? 

Here A=32 1,68; P=275,75; t=3.5 ; to find R, 
ftr Theor. III. 

From the Logarithm of A=3 2 1 ,68=2 . 50742 1 
Subftradtthe Log. of. . . F=275,75i=2.4405i5 

The Difference is the Log. of R« 0.066906 

which divide by the Time ... ^Z^S 

TheProdudlistheLog. of R=i.Q45=q.oi9ii6 

Then as i : 0,045 :: 100 : 4,5=4^/. pr Cent, the 
Rate required. 

Qucft. IV. InwhatTimewill275/. 15/. raife a Stock 
of 3 2 1 /. 13 s.'ffd. at the rafe pf 4il.per Cent. 
Compound Interpft ? 

Here P=275,75 ; A=32i,68; R=?i.p45; tp find 
t, per Theor. IV. 

. From the l^oe^ithm of . , A=:3?i,68=:2.50742i 
Subduft the Logarithm of 1*=:275,75=:2.4405I5 

J'^e P'lff. is the Log. of ... . Rt=i,p4^4=:p.66996 

XbentheLpg. pf 1.045=50.01511 6)0.066906(3, 5=:t. 
'..;..:' . 57348 

'— 955S0 
95580 



Thus the Anfweris 3 Years and 6 Months. 

iV#/tfi As the Amount of any Sum andf its Intereft is 
greater at Cemppund Intereft than at Simple Int^eft 
for any fitne above a Year j fo it is /^ at Com- 



J 



Cbap. VI. ^;^ Logarithms. j^^ 

pound than Simple Intereft for any 5iW Ufs than a 
Year, a5 the Learner may eafily ptovtf by tile 
Theorems beforegoing. 

5. Cy Annuities, £s?r. in Arrears. 

Theor. I. r~— = A = the Amount of any Sum. 

Thtor. 11. ^Cli = U = the Annuity^ Penfion^ 
&c. , 

Theor. III. ^^^^^=^ = Rt == the r/)^. 

; Thcor.IV. ^ «g-,R — R^ = thc Amount 

ofi/. 

6. Queft. L If 30/. yearly Rent be forborn 9 Years, 
what win It amount to at 5 /. per CenL Scc^ 
G>mpoqnd Intereft ? 

Here U=^o, t=9, R:5=i ^05 •, to find A ; per 
Theor. 1. 

The fcogapthin of , . . . R=i,05=o.o2 1 1^^ 
multiply by tlic Index ... g 

The Produft is . . . '. '. *. [ R*=i.5i6=ro. 1^0761 
To which add tfie'Ldg: of. . . 11=30=1 .477 1 2 1 

The Sum isthc Log. of UR'=45,48=i.657822 

Subduft.... U=36 

* I ' 

There reinainis . . . UR*— rU=i 5,48=1 . 1 8977 1 
Subftra«a the Log. of K— 1=0.05=1.8. 6g8gyo 

Tbcpiif.isthej UR'-rU . ^ ^ „ 

Log,of rT^ = A = 309,6=4.490801 

The Amount therefore is 309 /. 12;. foj- Anfwer. 

Que(t. 



142 Compmmd Inter eji 

Qjiqft. n, Whjtt Annqity, iSc. wfll ftiifc a Slock (or 
•mount to) 309/. lis. being forbom or un- 
paid 9 Years, at 5 /. ftr Cent. &c. Compound 
Intereft? 

Here A=:309,6 •, t=^ ; R=i,05 } to find U, per 
Theof. II. 

A^ S the Logarithm of ... . A=309,&=:2 .490801 
^ I the Logarithiit of . . . Rs=i ,05=0.02 1 1 89 

The Sum is the Log. of . . AR=s^5,q8^2.5i 1990 

Therefore the Log. of AR — A==i5,48=:i.i8977i 
Subduift the Log. of". . . R'^—i=xa.5 162=19.7 12650 

TheDifF.isthej AR->A ^^ ...^771177 
Lofrof f • • • TKCT =U = 30==i.477«i 

Th^t.is. 3pA /ff .^94p^i& the Jnnuity ipu^t. 

Queft. HI.. In what time will 30/. fer Annum a- 
•mount to '30^/1 T2J. ztgh per Cent. Com- 
pound Intfereft ? 
Here U— 3P ; Ai£309,6 1 Rdtr 1,05 5f to^d t, ^fr 
Theor.m. • •• / V 

^tEe Logarithm of . . . • A=309,6==2 .49.0801 
*Tthe Logaifthm of] ^; ; .'. Rz=i,o52=6.02ii89 

The Sum i& the Lpgr of AR=?3Z5,98=;? .511 990 

Subftraift .... ^ . A— U=279,6 

There remahir "^R.^U.^A=45,48=i.657822 
Sufcttraft the Ldgr of - * • U= 30=1.47712 1 

The^DHF. is the Log. of . . R'=:ivQ5^=o.i8o7oi 
Then the l^og^of i,(at5Q.02;;5i i$9>0ii8o7Oi(9=st. 
■ "' ^ I . 180701 

•I > ■ 
AriC\Ycr, 9 Yeys. ..... 

. ^. " Queft* 



Chap.VL 4y '-'^^^^^'^"Ms. m 

Qucft. IV. At what Rate per Cent. Compoimd In- 
tercft will 30/. Yearly Rent amount to 
309A i2s. in 9 Years? 

Here U=30 j A=309,6 ; t±=:9 ; to find R, pet 
ThconlV. 

From the Log. of . . . A— U:=:2 79,6=52446537 
Subduft the Logarithm of . • . 11=303:1.477121 

A— U 
The DifF. is the Log. of --^p =9,32=0.96941^ 

Agsun from the Log. of . . . A=309,6=2.49o8oi 
3ubftraft the Lc^^thm <^. . 17=30=1.477121 

Thel^. Isf he Log. of ^ =10,32=1.013680 

Therefore the faid Theorem IV. is ftduced to this 
Equation, viz. 9.32=io,32R — R\ Now this E- 
quation may be eafily refolv'd by a converging Series j 
or ftill eafier by the Tables of the Amount of i L 
for femoral Years fucceffivcly ; for therein R% or the 
Amount of i /. in 9 Years may be tried for the fevc- 
nd Rates p^r Cent, in the Tables. For inftancc, fup- 
pofe I pitch upon the Table at 5 1, per Cent. Com- 
pound Intereft, then I find R'=i.55i328, fcfr. 
Wherefore 9.324'i*55^3*8=io.87i328=io.32R ; 
therefore 10,32) 10.871328 (=1,05. Now fince 
i7. : 0,05/. :: 100/. : 5/. the very Ratep^r Cent, af- 
fumcd in the Tables, it will follow, that is the true 
Rate fought. 

Note^ This Qgcftion may be refolved by theRul^ 
of falfe Pofition 5 which I leave to the Leamcr^s 
Ewrcife. 



144' Compound Intefejl 

» 
7. Of the Present Worth of Annuities, 
tf r, at Compound Inter^. 

Here P denotes the prefent Wortf} of any Annuity, 
Leafe, 6f r. and the reft of the Letters as before. 
Art. 6. IThen the Theorems are as follows. 

Thcor. I. ..J— iL =: P = the prefent Worth. 
Theor. II. — ^^^ = U = the Annuity^ &c. 
Thcpn III. t>4.ji,^R = R' = the ^ime. 

Thcor. IV. ^ =^ R^ + R'—RS-?=: the Value 
ofR. 

Queft, \ . What is 30 /• per Annum to continue 9 Years, 
worth in ready Money, abating 5/. per Cent. 
&c. Compound Intereft to the Purchafer ? 
HcreU=30, t=9 j R=i,05 5 to findP, per 

Theor. I. 
The Logarithm of ... . R=:ij05=o.02 1 189 

Multiply by the Time .... 9 

The Prod, is the Log. of R5'=i.5i6=o. 180701 
which fubftraA from the Log. of U=30=r.477i2i 

»* ■ ' " ■ ■ , ■ 

The DifF. is the Log. of j*^ =i9,'788=:i.29642ro 
which take from ... 11=30 



there remains ... U — j;;^ =10.2 1 2 =1 .0091 1 1 
Subdudthe Log. of . . . R— i=o.05=.8.698970 

• The DifF. is the Log. of P=:204,24=2.3ioi4i 
The prefent Worth is 204/. 4^. 9 J. I, theAnfwer. 

Queft. 



I 



Chap; VI- hy LbGARitftMS. 145 

Queft. li. What Annuity, t£c. to continue 9 Years 
may one purchafe for 204/. 4/. gJi. abating 
5/. fer Cent. &c. to the Purchafcr ? 

Here P=204,24 ; t~^ \ R=i,05 ; to find \5if& 
Theor. II. 

r the Logarithm of .. ; . R^r= 1.5 i6z=:0. 180761 

Add ^ the LogaHthm of . . . . R=i,05=:O.C2ii89 

tthc Logarithm of . • P=2d4,24£ss2.3ioi4i 

The Sum i^ the Log. of PRR*:it:325, 11=2.512031 
Then from . . . • PRR*=325i,ii 
Subftrad ; : . . ; PR*=i:309.628 

there remains PRR'—PR*=i5,482=i. 189799 
Subduathe Log. of R'— i=:o.5i6 = .9.712656 

The Diff. is the Log. of ; . . Ue=30==i.477i49 
Therefore the Annuity fought is %oLprAnnumi. 

Queft. III. What tirhe may one enjoy an Annuity^ 
fcfr; ef ^ohper Annum;^ for 204/. 4 J. ^dh 
ready Money, abating to the Purchafcr 5/^ 
per Cent. dccJ 

Here P:^204,24 ; 17=^30 i R=j:i.d5 i t^ ^^ ^^ 
per Theor III. 

aaaS ^^^ Logarithm of . . Pfe2d4,24=2,3ioi4i 
-^^^ithe Logarithm of R=ii,05i=d.02ii89 

the Sum is the Log. of Pk=i=2 14,45^^2.33 1336 

Then... .i P±U=i234.24 

The DlfF. . . . P4.lJ^PR=£:ti9.788===i.2964t)2 
ifrhich fdbftl^a from the Log; of U==3d=i.477i2i 

thcDiff.tstheLog,6fRH=:l,o5^=i,5i6=o.i8o7i9 

U Then 



146 , Compound Inter efl 

Then the Log. of 1,05=0.02 n 89)0. i8o7i9(=9=it. 

I 8070 I ^ 

Anfwcr, 9 Years. 18 

Queft. IV. Suppofe an Annuity, t^c. t>f 30/. per 
Annum^ to continue 9 Years, be fold for 
204/. 4.5. gdi. yready Money, what rate of 
Intereft hath the Purchafcr allowed per Cent. 
&c. for his Money ?* 

Here P=204,24 ; U=3o •, t~9 ; to find R, per 
Theor.IV. 

From the Log. of the Annuity U:==: 30^1 .477 12 1 

Subduft the Log. of . . . . P— 204,24=22.310141 

-•■ • 

The Diff. is the Log. of -p =o.i4579=:.9. 166980 

Then the faid Theor.lV. is reduced to this Equa- 
tion, 'y/z. 0.14679=^0. i4679R5-f.R^ — R'^5 which 
is to b^ folv'd by the Method of Infinite Series. But for 
thofe whounderftand not that Method, this Queftion. 
h much better anfwer*d by the Rule of Falfe j or yet 
eafier by the Tables of Compound Intereft. See a 
compleat Set, with their Ufes, in my Syjiem ofDeci- 
fj$al Jrithmetic^ publiQied for Mr. Noon^ at the IVbiie 
Hart^ mCbeapfidey London. 

8. OfPurcbaJing Freehold Estates. 

FJtates in' Fee Simple (yjhkh are fuch as we com-^ 
monly call Freehold or Real Eftates) being pur* 
chafed Jor ^ver^ or without Reverjion \ 'tis plain, 
that in the foregoing Theorems for finding the pre* 
fent Worth of Annuities, (^€. if thofe Terms 
wherein t (= the Time) is found, be made to va- 
niih» aa being Infinite ; the faid Theorems will be 

reduced 



Chap. VI. hy LoGARiTftMS.^ f47 

reduced to fuch as fuit tht prefent Cafe^ and arc 
as follows. 

Theor. I. ^^ = P, the PrefintfFortb^ or Purchafe^ 

Money. 
Theor. 11. PR — P=U, the Armuiiyy or Efiate per 

Annum. 

^hcor. III. .^ =: R, the Amount of i/. at the 
given Rate. 

Quefiion. What muft be given for a Freehold Eftatc 
of soLper Annum^ allowing the ^uycr 61. 
per Cent. Compound Intereft for his Money ? 

fiere U^soy R=i,o6 5 to find P, p^r Theor. 1. * 

The Logarithm of . . . . 11-50=1.698970 

Subduft the Log, of R4-i=rO,G6=.8.778i5i 

The piiE is theLog. of ^ =P=833,j-=2,92o8 19 

< The Purchafe- Money, then, is 833/. 6s. id. If 
\t had been atthie. Race q( sLper Cent. Compmtnd 
In^r^Y the faicl Eftate would be worth lOOoA pre- 
fent Money i which is equal to' twenty times the 
Yearly Renf: ^hd therefore the* general Method of 
buying Freehold Eftates^ v& by paying 20 or 25 times 
the Yearly Rent ; which is commonly called 2q or 
25 Years Purchafe. 

The other two Theorems are wrought in the fame 
Manner for U and R j which, being fo very eafy, 
«ieed no ^.xa^iple. 

U 2 CHAP. 



148 Vulgar FMStiQm 

CHAP. VII. 
Cy Vulgar Fractions ^jr Logarithms* 



i.fTp 



O find the LogarUbm of a Vulvar Fraffion^ 
this is the 

iFrom the LDgarithm of tht J^TumerAtai^ 
Subftraft the Logarithm of the Denominator 5 
the Remainder is the Log. of the Fraftion. .^ 

Exampl? I. What i$ the Log. of the FradUon ^3 ? 

From the Logarithm of . . . 5^^^-707570 

Subftiad the Logarithm of . • . 73=1,863323 

there remains the Log. of H'=o.69862==.9.844247 

Exam. II. What is the Logarithna of 77^5-6 ? 

From the Logarithm of ... . 13=0.000000 

Subftra£t the Logarithm of . • . 1 7562=3.244524 

there remains the I , _ rsrs^r^r,.^— #c *,^^^^< 
Log. of r • "^V^ - ><^o^56947=T.P/755i76 

Exam. III. What is the Logarithm of r^, > 

From the Logarithm of ... . 100=2.000000 
Take the Logarithm of . . , 1 357= 3 . 1 32589 

there remains the Log of-^==,073692=.fe.867420 



Exam. IV. What is the Logarithm of r^ 



eo ' 



? 



T¥ 



C^p.VII. ^y Logarithms.' 149 

The Logarithm of ; . . . 5973=3776192 

From which take the Log. of ioooooo=6.poopoo 

there rem. the Log- ofT^^=,oo5g73=.7.776i9^ 

2. If the Fraftion be a mi^done^ it muft be re- 
duced to an improper FraSion \ apd then procce4 
wit|i it as before. 

Example L What is the Logarithm of 13I ? 

fiere I3f=f 5 

Therefore from the Logarithm of 96=1.982271 
take the Logarithm of . . • . 7=0.845098 

there remains the Log. of 131=13.714=1.137173 

Exam. II. What is the Logarithm of 193^ ? 
Here 193,4=1^; 

Then from the Logarithm of 14140=4,150449 
take the Logarithm of .... . 73=1.863323 

fhererem.tl^c Log, of I937i=i93>^986?=2.287i26 

3. If the m\x'd Fraiiion confifts of large Num^ 
hefSj it may be moft eafily reducM by Logarithms, 
thus 5 fuppofing the Example be 2145^*-^ j 

To thp Log. of th€/«/^jfra/Piir/ 2145=3.331427 
iVdd the Log[. of the Dehominator 589=2.7701 15 

The Sum is the Log- of . , 12^3405=6.10154:2 . 
to which add the Numerator . . 57 

ithe new Num. 1 263462 7 .. t.^^ 
theDenomin. S7 3^^"SW. 
* . its *' • 

4< 



t^o Fui^ar FraStions 

4. Ta mtdfipfy Vulgiir Fraffions hj Logarithtnsi^ 
tdd the Logarithms of the Numerators for the Lo- 
garithm 0f a new Numerator ; and the Logarithms of 
the Denominators for the Logarithm of a new De- 
nominator* 

Example. What is the Prbduftof -j^^ into -J ? 

AddtheLog«.oftheNumerators|' 35=;;|440^^ 

The Log. of the new Nuinerai»r 245=^2.389166 

Add the Log., of the Denominat'. j '^P^'^f^^t^ 

The Log. of the ne'iv l)enoinin. i557=;=:3.i9228S 

Therefore ^ x J =c: ifti^ the fradional Produft 
required. 

5. The Logarithm oftlfeProdu^ offevera! Frac- 
tions muitipiied into one another is thus obtained ^ 
c/2.- Add the Logarithms of all the Numerators and 
the ^tbmetwai Complements of the Logarithms of 
all the Denominators together ; the Sum is the Loga- 
rithm required. 

^M^ What is the Logarithm of ^-^ :?c H x t ? 

-; • 3 the ]^pgirithms of , < 33=1.518514, 
Addto-J^ 'I 3===o.477i2^ 

l^^^-.WAnih Co.p .^the ^^:^ 



Chap.Vit 1^;^ Logarithms, 151 

€. To divide Vuigar Fradiohs by Logarithms^ 
do thus; Add tlic Logarithm oF the Denominator of 
the Divifor to the Logarithm of the Numeratvr of 
. the Dividend \ the Sum is the Logarithm of a iHVf 
Numerator \ and the Sum of the Logarithms of the 
Other two FaSors^ is the Logarithm of the new De- 
nominator of the Quoticat required. 

Example! Divide ^ by h 

AddtheLogaritkmsofj .^'S^f^^S 

Thfe Log. of the new Numerator 2205=3.343408 

Add th. Logarithm, off . . . ,J^ g;^^g|| 

The Log, of the new Denom. 10899=4.037386 

therefore i).^,(«^^,=^. See Art. 4. 

7. The Logarithm of this Quotient tnay be found 
by one Addition, in like manner as dittfted in Art. 5^ 
thus ; 

gether j the Arith.Comp. ofc 7=.9. 154902 
t the Logarithms of I i557=.6.8o77ia 

The Log. of . , i)rrf;(=:^,=,2023i=:.9.3o6o22 
B 

f 8. The Extraaion of the 2?i9(?/j of VuJf^ar Frac- 

I » tions by Logarithms is thus performed. Divide the 

Logarithm both of the Numerator and Denomihatot 
of the given Fraftioti by the Inde:^ of the Root ; the 
^otients Ihall be the Logarithms of the Numerator 
and Denominator of the FraSional Root required.' 

I . Examp. 



152 Fiitgar FradHont 

£xamp. I. What is the Square Root of the Fraftiofi 

SOlOl * 

Divide the Logarithm of ..... 1849=3.266937 
by the Index of the Root ... 2 



The Log. is the Num. of the Root 43=1.633468 

Again, divide the Log. of. . . 10261=4.008643 
by . 5 i . ; . 2 



ThfeLog. of the Uenom. of the Root 101=2.00432 1 
Therefore v"^^ sc 1^ the Rodt required. 

Example IL What is the Cube Root of 86*-|i .? 

This reduced to an impropcf Fradliorii Is ^f^' j i 

Therefore the Logarithm of 29791=4.474085 
i thereof is the Logi of the new Num. 3 1= i .49 1 3 6 2 \ 

Again, the Logarithm of . . .". 343=2.535294 
T of which is the lx)g. of the »^wZ)^»^w. 7=0.845098 

Confequently ^86j|f = 't = 4^, the Cube Root i 

fought* 

^ 9. To find the Logarithm of the Root of any Frac- \ 

tion ; add the Logarithm of the Numerator to the 
Jritbmetical Complemehi of the Logarithm of fhe 
Denominator^ and divide that Sum by the Index of 
the Root \ the Quotient fhall be the Logaritfim 
fought. ^ . 

Example L What is the Logarithm of the Square 
Root of theFraftion -^'i ? 

the 



f 

> 



Chap. Vil. iy Logarithms^ i^j 

The tx)g. of the Numerator . . • 1849=3.266937 
Gomp. Arith. of the Log. \ ,^^^__ ^ 
of the Denom. } • • I020«=.5.99i357 

The Sum is .9.258284 

which divide by the Index of the Root 2 

The Quot; is I y-trrs- *^ ^r 

theL:og.of } /t.^.=*1V.«42574=.9.629I47 

Exam, IF; What is the Lb^rilhih of the Cube Rooi 
of the mix'd Fhuilion 863*11, or its equal 

»079i i 

A jj Cthe LogaPithhlof*. . ; ; 29791=4,474685 
\ the Ar. Com^ of the Log. of 343=7.464706 

The Sum is the Log. of .... . 681^1=1.93879* 
^hich divide by the Index of the Root . • 3 

c ii A P. viiL 

Duodecimal k^iT'^w^tiQ perfirmed by 

LbOAfelTHMS. 

i. Qj I N CE this kind of Arithmetic is fo very co)n- 
\j mon^ and yet in the common Way fd very dif- 
ficult ; I hdpe 'twill not be unacceptable to the yeung 
Artificer to be convinced with bow much more Eile 
land Pleafure he may compute his Dimcnfions in this 
Way by the help of Logarithms. And as it is pro- 
p&T to reduce them firft to Decimals^ I have here fub- 
. . X joined 



'54 



Duodecimal Arithmetic 



joined a Table (hewing by InfpeAion the Decimal 
Parts of a Foot (in this Cafe the Integer) anfwering 
to any Number of Primes^ Seconds^ and Thirds -^ 
which are the Parts or Diviiions of a Foot made ufe 
of in this kind of Menfuration. 



-^2. 



The Duodecitnal Table. 



Duode- 
cimals 



I 

2 

3 
4 

5 
6 

7 
8 

9 

10 

II 



Decimal Parts. 



Primes ' ISecondsl Thirds 



»o8j333 
^1^6666 

J33333 
,41^666 

5 

>583'333 
,^66666 

75 

^^213333 
,916666 



,006944 

013^88 

,02o83'3 

,02^777 

.0347A 
,041^66 

,048611 

og5555 
,0625 

,069444 



n/// 



000578 
,001157 
,001736 
,002314 

002893 
,003472 
,004051 
,004629 
^005208 

005787 
,006365 



3. The foregoing Table is too eafy to need De* 
icription, I mean for any Perfon concerned in Duo-^ 
decimal Menfuration ; and therefore I ihall proceed 
to exemplify and illuftrate the Operations this Way 
by Logarithms in all the Rules of Multiplication^ 
Divifiotij Involution^ and Extraction of Roots. 

4. Mutiplication ^2/* Duodecimals by Loga- 
rithms. 

Examp. I. Suppofe a Plane be 9^ 10' in Length, and 

8^ 8' in Breadth v ^^re the Content or Area ? 

^, , c the Log. . . of 9^ io\=9. 83^3 3=0.99 2 704 

•^^^itheLog.of.. 8f. 8'.=8.6666=o.937849. 

The Sum is the Log. of . • . . 85,;2f22asi.93055g: 
Thatis, 85,222 Feet,=:85^ 2'. 8^ the Area requirecL 

Examp 



Chap.Vli. ^Logarithms. 155 

Examp. II. What is the Produft of 40'. 9'. 10* by 

ii'.9'? 
- , , c the L. of 40f. 9'. io'.s=40.8 194= 1.6 10866 
-^^"l the Log. of 11'. 9'=o,979i6=!.9.99o854 

The Sam is the Log. of ... . 39,969=1.601720 

Therefore 39,969^39^. 11'. y'.S'". the ProduS, 
or Area required. 

Examp. in. What is r 75 Feet 00'. 04'. by 8'"? 

. , . 5 the L. of 175'. o'. 4*.=i75,o2^=2.243io7 
^^** I the Log. of 8"'.=,oo4629=. 7.665487 . 

The Sran is the Log. of . . ,8/1 1 1 =.9.908594 

Therefore o,8InI=o^ 9'. 8*. 9"' = the -rfr^« 
fought. 

Examp. IV. What is 17'. 9'. 2". &" by 6'? 

.,,jtheL.ofi/.9'.a*.6'"=i7.76736=i.2474i.7 
^'***i the Log. of .... 6 ^0778151 

The Sum is the Log. of . . 106,60416=2.025568 
Thusio6.6o4i6=io6f. 7' 3'. the Area fought. 

Examp. V. What Number of 7&/f^ Feet is in a Ctllar 
2if. 2'. long, I If. 10'. 8'. broad, aad 7f. 3'. 
deep? 

r the Log. of. • »»'• 2'=2i>i06=i.325659 

AddStheLog.of iif. 10'. 8'=ii.?88=i.o74io9 

ttheLog.of 7'- S'- =7»g5' =0-«^0 338 

The Sum is the Log. of . . . 1820,19=3.260106 
Therefore i820,i9=i820f. 2'. 3'. 4'", the SelidHy 
required. 

X 2 . 5- 



1 56 Duodecimal jirithmetic 

5. Division ^/Duodecimals^ Logarithm^. 

pxamp. I. What is 85^. 2'. 8^ divide by 8^. 8' ? 

The Log. pf • . . . Sgf. 2{. 8^=:85,;2f2=i. 930553 
SubduA the Log. of 8^. 8' =8,i|?66=o.937849 



rr 



The DiflF. is the Log. of . . . 9 -83^3 3=0.992 704 
Sothat9.83'33=9^. 10', theAufwer. 

Examp. 11. What is 9'. 8^ 9^ divided by 8''^? 

From the Log. of 9^ 8^ 9"'=p.8xiii=.9.9p8594 
Subftraft the Log, of 8'"=,oo4629=.7.665487 

The Diff. is the I^og. of . . .. 175,02^=2.243107 
Therefore the Anfwcr is 1 75,02;/=! 75f. o'. 4.". 

Examp. in. Divide 39^ 1 1'. 7^ &". by 40*: 9'. lo^ ' 

^Loltf }• • 39^- XI'. 7^. 6^''=39.9687=i.6oi720 
Subd. the Log. of 40^. 9'. 10^=40.8194=1.610866 

The DifF. is the Log. . . of pr979i6=.9.99p854 
But ,97916=1 1'. 9^. the Quotient required. ] 

Examp. IV. pivide 106^. 7'. 3^. by 6,. 

From the L. of io6f 7'. 3^,=io6,6o4i^2. 205568 
Subftraft the Log. of 6=0.8 78 1 5 1 

* The DiiF. is the Log. of 17,76736=1 .247^17 

Jhus 1 7,76736=1 7^. 9'. 2". 6"'* the Quotient 
fought. 



^. 






chap. VIII. /^Logarithms. 157 

0. I NVOLuTiON pf Duodecimals iy Loga* 

RITHMS. 

Examp. I. What is the Area of that Square whofe 
Side is 12^. 9'. 7^ iq'^'? 

TheLog.of 12'. 9' 7^. xo'^=i2.8p439=s:i,io7359 
Multiply by . • • 2 

The Produft bthe Lc^. of 163.9524=2.214718 

Therefore 1 63.9524=1 63^. u^ 5^ i''>. the ^r^tf 
required. 

Examp. II. What is the Solidity of a Cuh whofe 

Sideisi^2'. 9^ ii''^? 

The Log. of if. 2'. 9^. I i''^=i,23553=;o.09i854 
which multiply by , . . ^ 

The Produft is the Log. of 1,877/1=30,273562 

Therefore f,8774=i<'. ip^ ^^. 4^ the Solidity 
fought. 

7. Extraction (J^Roots <?/ Duodecimals 
^Logarithms. 

Examp. I. What is the Side of that Square whofe 
Areai8i63f. 11'. gf. i^^i 

The L.of 163. 11'. sf. i'''.=i63.9524=2.2i47i8 
Divide by . . . 2 

The Quotient is the Log. of 12.804392=1.107359 
Thus 12,80439=1^^- 9'- 7^- 10'^'. the Side fought. 

Examp. II. What is the SiJe of that Cuk, whofe 
Solidify is if. i&.6^.4"'i 

The 



t^S Commm Ruksofjiriibmetie 

The Log. of X. lo'. 6^. 4'".=i.S774=20.a735(Jft 
Divide by ... . 3 

The Quotient is the Log. of I.23553=:0.09i854 

. But 1,23553=1^. 2'. /. ii'^ the Side of the Cube 
required. 

Thefe few Examples abundantly (hew with how 
much more Eafe^ Brevity and Expedition the Opera* 
tions of Duodecimals are performed by Logarithms^ 
than by the ordinary Method. 

CHAP. IX. 

Ti&^ OpfeRATiON of the common Rules ^A- 
RiTHMETic ^j^ Instruments; viz. the 
Logarithmic Scale ; and Gunter'^ 
Line, noitb the Compasses, and on the 
Sliding-Rule. 

I. TTAving in the fbeory /hewn the Nature and 
Jl X ConftruBion of the Logariibmic Scakt and 
hunter's Line 5 I (hall here briefly cxAplify their 
Ufes in the Operation of the common Kuks of A- 
rithmetic thereby; and in doing of this I (hall ob- 
ferve this Method ; firft, to give the Operation by 
Logarithms in Numbers. Secondly,^ to perform the 
fame by the Logarithmic Scale. Thirdly, to work 
the fame Cafe on the Gunter with the Compajfes ; and 
fourthly, to do the fame thing on tht Sliding-^Gunter. 
In this Method, the Analogy or ^reement between 
the Numerical ahd Iifftrwnentat Operations will more 
eafily appear; and the Nature and Reafon of the 
latter be much better underftood hy young Learners. 

2^ 



Chap. IX. hy iNSTfttJUXNTs. 159 

a. Multiplication^. 

Examp.!. Multiply 9^7* 
Firft, by Logarithms. 

^ J , 5 the Log^thni of 9=^.954241 

^^^ I the Logarithm of 7=0.845098 

The Sum is the Log. of the Prod. =63=1.799340 

3. Secondly, by the Logarithmic Scale. 
(Ifote. If the fmalkft Divrfions in the Line AD 

reprcfcnt Numbers^ the Logarithms begin from C c 
in the Line Ae } if the middle Divifions in A B be 
Numbers, the Logarithms begin from G 5 but if the 
hrgcft Divifions in AB be taken for Numbers, then 
the Logarithms begin from H, in the faid Line A e^ 
And fince xkz finaUejUhvifions are toofmall^ and the 
largefi Divifions too large for Examples, we muft 
coniequently chufe the mean Divifions in A B to rc-« 
prefent the Numbers 1, 2, 3, 4, (^e. or 10, 20, 30, 
40, 6fr. or lOo, 200, 300, &r. and fo the Loga^ 
rithms begin from G. Therefore) Set one Foot of 
the Compailes in G, and extend the other to the 
Logarithm of the Multiplier 7=ag, which you^l] 
find to be Ga=845, and fince G b=954 is the Lo^ 
garithm c£the Multiplicand 9=bh ; therefore with 
the fame Extent G a in the Compares, fet one Foot 
in b, the other will fall on d ; therefore Gd is the 
Logarithm of theProdud: dm=63 xn A B, the Num^ 
ber iought. 

4. Thirdly, by the Gunter with CompaflTes. 

Set one Foot of the Compafles in the Beginning of 
«iie Lmeat i, and extend the other to 7 ; with that 
Extent in the Compafles fet one Foot in 9, the other 
witt fikU Oft -^3, the Produ<3: fequiired^ 

NoiK 



i6o Commoh RuksofArithfHetid 

Note. When the Numbers zxtfmall the larger l)i-^ 
vifions may be ufed, as in the prefent Cafe ; but il 
the Numbers be large^ the lejfer Divifions muft be 
ufed. 

$. Fourthly^ by the Sliding-Guntef. 

In this Cafe, there is one Line of Numbers on the 
Rute^ and another on the Slider^ both mark'd with 
N, at the End. And it is eafy to conceive that by 
Aiding one of thefe by the other, the fame Effefts are 
poduced as before with the Compares ; that is, any 
Part of the Line on the Rule is transfer^d to, or com- 
pared with any other Part of the faid Line by means 
of the Aiding Line. 

Therefore fot i on the Slider to 7' in the Line on 
the Rule ; then againft 9 on the Slider, you find 63 
on the Rule, and that is the Produdb fougkt 

6. Examp.II. BytheG«/i/en WhatistheProdu^E 
ot27 by 18? 

Here the leAer Divifions mtift be ufed, aiid the 
greater ones reckoned 10, 20, 30, 6ff . ort thtfirjl 
Radius ; and confcquently on the fecoHd Raiiiis they 
will be 100, 200, 30o,.£s?f. if the double Radius be 
ufed. For then it will be 10 : 180 :: 27 : the Produdt 
fought. But fince 10 : ,180 :: i : 18 ; therefore if you 
make i : 18 :: 27 : the Produft 5 the Jingle Raaiui 
will give the Anfwer in the fame manner ; only re- 
membring that th^ fourth Number fought will be of 
the fame Denomination with the fecond^ which in this 
Cafe is Hundreds. 

Therefore fet one Foot of the Com{)a{les in the Be- 
ginning of the Line, and extend the other to 18, the 
ume Extent will reach from 27 to 486^ the Produ6fc 
fought. 

And by thtSliding'Rule^ thus; fet ion the Slider 
to 18 on the Rule, and then againft 27 on the Slider 

you 



I 



Chip. IX. 4y Instruments. . i^f 

you find 486 on the Rule, which is the fame Producft 
I as before. 

^ 7. Examp. III. What is the Produdt of 257 by 34 ? 

Take in your Compafles theDiftancc from i to 34 
on the Line of Numbers^ the fame Extent will reach 
from 257 to 8738-, thie Produft required. 

By the Sliding- Rule, thusi Set i on the Slider t^ 
34 on the Rule, and againil 257 on the Slider, yoa 
fee 8738 on the Rule^ which is the Produft as 
before. 

S. Examp. IV. What is the Produd; of 215 by 

108? 

With the Compafles, take the Diftance from i to 
ko8 on the 'Gunter^ the fame Extent of the G)m- 
pafles will reach from 215 to 23220, the Produft re* 
quired. 

By the Sliding Rule, thus \ Set i on the Slider to 
7 108 on the Rule, andagainft 215 on the Slider you 

find 23220 on the Rule, the Produdl fought. 

9. When the Produft becomes fb large, it muft 
be a v^ry large Line of Numbers indeed to (hew it 
near the Truth ; the Ufe of thefe Lines being prin- 
cipally where the Numbers arefmall-, or where great 
Exaftncfs is not required. They who underftand 
the foregoing Doftrine of Logarithms can never be 
at any lofs to know how many Places of Figures are 
contained in the Number fought, in this, or any of 
the following Rules. 

to. Division. 

Examp. I, What is the Quotient of 63 divided by 9 ? 
Firft, by Logarithms, 

y ' From 



1 62 CofHmon Rules of Arithmetic 

From the Logartthm of . . . 63=1.799340 

Subdud the Logarithm of . . . ' 9=0.954242 

The DiE is the Log. of the Quot. =7=0,845098 

tu Secondly^y by the Logarithmic Scale. 

From Gd=i799 the Logarithm of dma=63, take 
Gb=954 the Lc^arithm of bh=9 ; and there will 
temain Ga=845) the Logarithm of ag=7, the 
Quotient fought 

12. Thirdly, by Gunt&^s Line and CompafTes* 

Set one Foot of the Compafles in i, and extend 
the other to 9, and then with that Extent of the 
Compafles fet one Foot in 6^^ the other will fall (to- 
wards the beginning of the Line) on 7, the Quotient 
fought. 

13. Fourthly, by the Sliding-Rule. 

Becau(e 9 : 63 :: i : the Quotient ; therefore fet 9 
on the Slider to 63 on the Rule, and then againft i 
on the Slider is 7 on the Rule, which is the Quotient 
fought 

14- Examp. II. What is 486 divided by 18 ? 
By the Gtinter and Compafles. 
Extend the Compafles from i to 18, that Extent 
will reach from 486 (downward) to 2 7, the Quotient 
required. 

By the Sliding-Rule. 

Set 18 on the Slider to 486 on the Rule, then a-> 
gainft I on the Slider you find 27 on the Rule, the 
Quotient fought. 

All other Operations of Diviflon being performed 
in the very fame manner, 'tis needlefs to add any 
more Examples of this kind» 

.'5- 



Chap. IX. hy iNSTRUMENts. \h\ 

15. iNVOtUTJON. 

Examp.!. What is the Square of 9 ? 
Firft, by Logarithms. 

The Logarithm of . . . 9=0.954242 

Multiply by the Index . . • % 

The Prod, is the Log^ of the Square =8 1=1 ,908484 

iS. Secondly, by the Logarithmic Scale. 

Let the Logarithms begin from G in the Line 
A e, as before 5 then with the Compafles take thp 
Diftance gb=954 the Logarithm ofbhi=PB9; and 
with one Foot remaining in b, turn the Compafles, 
the other Foot will fall on n; then fliall Gn=a: 
(aGb»)i9o8 the Logarithm of n 0=81, whiclj 
therefore is the fquare Number fought 

17. Thirdly, by the G«»/^r and Compafles. 

Set one Foot in i, and extend the other to 9, wherq^ 
keep it fix'd, and turn the Compafles, the other 
• . Foot will fall on 8 1, the Square fought. 

18. Fourthly, by the Sliding-Rule. 

I Becaufe i : 9 :; 9 : the Square required, therefore 

fpt I on the Slider to 9 on the Rule, then againft .9 

' on the Slider is 8 1 on the Rule ; whi<?h is the/^uare 

Number (ought. 

19. Examp. II. What is the Cube of 9 ? 

t . By the Gunter and Compafles. 

'- Extend the Compafles from i to 9, that Extent 

will reach from 9 to 81, a^d agai^ from^ 8j to 729, 

the Cube Nunxber required. 

• Ya Br 



i 



1 64 Common Rula of Arithmetic 

. ■' . 

By the Sliding- Rule. 
Set I on the Slider to 9 on the Rule, thep againft 
9 on the Slider is 81 on the Rule,' and againft 81 on 
the Slider (remaining unmov'd) is 729 on the. Rule, 
the Cube Number required. 

20. Examp. III. What is the Square and Cube of 

the Number 37 ? 

By the Gunter and Compafles. 
Extend the Compafles from i to 37, that Extent 
•will reach from 37 to i '^6^ the Square \ and the fame 
Extent will reach from 1369 to 50653 the C«^^ 5 
both a3 required. 

By the Sliding-Rule. 

Set I pn the Slider to 37 on the Rule, then againft 
37 on the Slider is 1369 on the Rule, which is 
Square \ and againft 1369 on the Slider (remaining 
unmov*d) is S^^Si ^^ ^^e Rule, which is Cub^ of 
37 5 both as before. 

21. Extraction of Roots* 

Examp. I. What is the Square Root of 81 ? 
Firft, by Logarithms. 
The Logarithm of ... . ' 81=1.908484 

which divide by the Index • . . . z 



The Quotient is the Log. 7 ^ ^ «^.r. .^ 

ofti:.SquareRoot ^t ....=9=0.55424^ 

22. Secondly, by the Logarithmic Scale. 

Bifed Gn=i9o8 the Logarithm of no=8i, in b ; 
then (hall Gb=954 be the Logarithm of the Square 
Root, viz. bh=9, the Number fought. 

«3- 



chap. IX. 4y Instruments. -165 

23. . Thirdly, by the G«;«/^r and Compafles; 
Take with the Compafle? the Diftance between i 

and 81, and bifeft it; then take one Half in the 
Compafles, and it will reach from i to 9, th^fquare 
Root fought. 

24. Fourthly, by the Sliding-Rulc. 

Move the Slider forwards and backwards till you 
make the fame Number on the Rule anfwer i on the 
Slider, as on the Slider anfwers 8i on the Rule; 
which Number will be the fquare Root fought, and 
in the prefcnt Cafe will be found 9. 

25. Examp.^II. What is the Cube Root of 50653 ? 

Divide the Diftance between i and 5065^ into 3 
equal Parts ; the firft Divifion will fall on 27 j the 
Cuie Root required. 

- Noie^ In the double Line of Numbers, if the 
grand Divifions be efteem'dt7»i/j in the firft Radius, 
thofe in the fecond Radius will be Tens ; if thofe in 
the firft be Tens^ viz. 10, 20, 30, (^c. thofe of the 
fecond will be Thoufands^ as 1000, 2000, 3000, (^c. 
with regard to fquare Numbers : and confequently in 
- ExtraSion •, if the Number whofe fquare Root is 
fought be lefe than 100, yet greater than 10, the 
Number it felf will be found on tliQ fecond Radius i 
and its Root a Number of Units onthtfrft Radius. 
But if the Square be lefs than 10, both it felf and 
Root will be found in tht firjt Radius.' Again, if 
iht fquare l^umber h^'htVNG^n 1000 and loooo, the 
Number it felf will be found on the fecond Radiits^ 
and its Root a Number of Tens on the firft Radius^ 
But if it be between 100 and 1000, both the Num- 
ber and its Root of Tens will be found on th^firfi 
Radius. After the fame manner you may rcafon 
concirnbg the Cube Number and its Root. 



i66 Common 'Rules of Arithmetic 

26. Since the Logarithm of the Square h dduhl$ 
the Logarithm of the Root ; and the Logarithm of 
the Cube triple the Logarithm of the Root : therefore 
if a Line of Numbers of a ^ngle Radius, be equal to 
another of a double Radius^ and thefe two appofitely 
laid together, beginning from the fame Point ; thea 
againft any Number on th& Jingle Radius^ you fee its 
correfpondent Square on tht double Radius i andfuch 
Lines you have on fome Sliding- Rules. 

Alfo if a Line of Jingle Radius^ were made equal to 
another of a triple Radius, and thefe exa^ly and 
properly placed together, then the Numbers on the 
latter would be the Cubes of thofe on the foimer; 
and fo the Square and Cu^e Roots of ^ny Number j 
and vice verfd, would be difcoverable by Jn/peff ion. 

%"]. Moreover by means of ^ Jingle and double Line 
of 'Numbers made to Aide by each other, *tis very 
cafy to find a mean Proportional between any two 
given Numbers ; as fuppofe 13 and 23. .Thus; fet 
, 13 on the double Line to 13 on the Jingle one^ then 
againft 23 on the double Line is 17,35 on the Jingle 
cne, which is the Mean required between 13 and 23. 
Or if you fet 23 to 23, then againft 13 on the double 
you find 17,35 on t\iz Jingle Line, the A&<?» required 
9S before. 

2 8 . In like manner, by means of ^Jingle and a triple 
Line of Numbers, two mean Proportionals may be 
cafily found between any two Numbers, as 2 and 
54 ; thus ; fet 2 on one Line to 2 on the other, then 
againft 54 on the triple Line is 6 on the fingle one, 
which is the Jirjt Mean 9 then fet 2 on the triple 
Line to 6 on tht^ngle one, and againft 54 is 1 8, the 
fecond Mean, on the fingle Line ; fo the four Num- 
bers are 2 : 6 :: 18 : 54. And thus you may find 
two Means between any other two Numbers, which 
-in many Cafes is a moft ufeful Problem. 

29. In thf foregoing Qperation? } have^made no 
jl)enti9n oi Decimals, becaufe they are to be refpeded 



Chap. IX. ^;^ Instruments* 167 

as Whole Numbers in the Management of them by 
Inftrumnial Operations ^ in the fame manner as they 
yrtrt by Numerical Logarithms i the Number of^-> 
£imal Places in any ProduSl^ ^oHents Power^ Root^ 
&c. being always determined here, as in all other 
Methods of working them, by the Rules proper to 
Decimal Arithmetic. 

30. Thus it appears what Similarity^ CobereHei,^ 
and mutual Relation there is between the foregoing 
Methods of folvlng Arithmetical ^eftions by Loga- 
rithms, both Numerical and Inftrumental ; and that 
they are all one in Nature, and differ only in the 
Modus operandi^ or Manner and Form of working. 
By this Chapter, I prefume, 'twill be eafy for the 
Learner to obferve how any common Queftion in 
Arithmetic^ or the Menfuration of Artificers Work^ 
as Joinery^ Mafonry^ Carpentry^ Paintings Ttmber- 
MeafurCj Gauging ^ &c. may be moft readily per- 
formed by the Line of NumberSy with Compaflcs, or 
by the Sliding- Rule ^ which is much the beft Way. 

CHAP. X. 

Sbeweth the Analogies ^r pRopoRTioNsy^r 
the Solution of all (he Cases g^ Plain and 
Spherical Triangles, both Right and Ob^ 
[ liquc angled. 

1, TF any defire to be throughly inftrudcd ih the 
X ^eory of Plain and Spherical Trigonometry^ I 
muft refer them to my Toung Trigonometer^s Guide ; 
fince all I intend here is only to ftiew ihegreal and 
moft excellent Vfe of Logarithms in the praftical Re- 
fojution of Plain and Spherical Triangles. A Sy- 

nopHs 



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ain Tri- 



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chap. X. and Spherical Tkiaj^gles. * r69i 

2. A Synopfifi of the Analogies for the Solution of 
all the Cales of Oblique-angled plaiii Triangles. 



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In the firft Triangle, 'tis AD— DB l . p 

in the fecond Triangle, AD+DB 5 

Then AB — AG i= GB = 2DB. Thus each Tri- 
angle is reduced to two Right ones, viz. ADC, 
and BDC ; in either of which two Sides are 
known whence the Angle C may be eafily found. 


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sC : AB :: sA : BC. 




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yjo Solution ofPUin 

« A Synopfis of the Canons^ and the Analones (formed firom them 
Ae Solution of a^Jpfte Cafes of Right-angled Spherical Triangles. 






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172 SolutiohofPIainTKiAViQhEs. 

5. Being thus fiirniflied with Proportions, we 
ihall foo|i fee with what incomparable Pleafure and 
Eafe the fcveral Cafes of Triangles before going are 
refolved by the Canon of Logarithmical Sines^ Tan* 
genfSf and Secants ; and alfo by the Line ofl^umbers^ 
both with the Compajfes and by the Sliding^Rule ; I 
fey, we ftiall fee in the two next Chapters, with 
how much more Eafe and Pleafure they are re- 
folved by thefe two Methods^ above what is attain^ 
able by any other Way yet invented for this Pur« 
pofc. 




CHAP, 



Chap. XL Solutioff of Plain Triangles. 173 

CHAP. XL 

5^^ Solution ^Plain Triangles ly the 
Canon ^Logarithmical Sines and 
Tangents ; iy Gunter's Scale and 
Compasses s and ^ the Sliding Rule. 

X. T Have already defcribed and taught the Ufe of 
J^ theXogarithmic Canon, fo far as to find the 
I^ogarithms of any Number j Sine^ Tangent ^ qx Secant 
propofed, I fliall therefore here only obferve, that 
what I call Gunter^s Scale is fuch a Plane Scale as hath 
upon it Gunner's Une of Numbers^ and of Artificial 
Sines and Tangents j whofe Nature, Conftrudipn, 
and Defign, have been before difcourfed of in the 
Theory. The Sliding-Rule has the fame Lines, 
which are contrived to Aide by one another as you 
pleafe ; and to avoid Repetitions, I fli^U call the 
Line of Numbers on the Rule it felf, A ; and that 
on the Slider, B ; alfo I fhall call the Lines of Sines 
and Tangents on the Rule Sr, Tr ; and thofe on the 
Slider Ss, and Ts. You muft know alfo that the 
End of each Line marked 10, 90, 45, is here called 
Radius. Having premifed thefe things, we proccei 
immediately to the Solution of 

2. Right-angled Plain Triangles, 

C^fe L In the Right-angled Triangle ABC, there is 
given the Bafe A3=285, and the Angle a$ 
Bafe B=:32^ 48' 5 to find the Perpendicular 
AC, and the Hypothenufe BC. The Hy* 
poth. Radius, Fig. I. 

The Analogy for AC is, sC : AB :; sB : AC. 



174 So/uffon of Plain Triaijglies. ' 

In Words ; 
As thcSincof the Angle C=57° ii^^g.gij^gyz 

h to the given Side or Bafe AB=2 8 5=2 .454845 1 , • 
SoistheSmeofthe Ang, 3=32^ 48^=9.733765 J^^^ 

12.18S610 



To the Perpendicular AC=i 83,67=2.264038 

The Analogy for EC is, sB : AC :: R : BC 

In Words ; 
Asthe Sine of , • . B=32^ 48'=9.733765 

is to the Perpendicular AC=i 83,67=2.264038 
So is Radius • . . . 90^ oo'=io.oooooo 

to the Hypothenufe. . . BC=339,o6=2.530273 

^. By Scale and Compajfes. To find AC 

Set one Foot of the G)nipaflres to 32^ 48'=B, in 
Line of Sines, and extend the other upwards to 
57^ i2'=5:C 5 the fame Extent will reach from ^^5 
s=AB downwards in thpLinc of Numbers to 183,67 
=sAC, the Perpendicular fought. 

To find BC. 

Extend the CompaiTes from 32^ 48'=B to Radius 
90^ in the Line of Sines ; the fame Extent will reach 
(in the Line of Numbers) from i83,67=AC, to 
339,o6;=:BC, the Hypothenufe fought, 

4. By}he SMng'Rule. "to find AC. 

Make the IJne of Numbers Aide by the Line of 

y Numbers, then will the Lines of Sines Aide by each 

other, and alfothe Line of Tangents by the Line of 

Tangents. Having thus prepared the Rule; fet 



Chap. XI. Solution of PlainTKiAnGLn^l iyj^ 

57^ 12' on Ss to 32^ 48' on Sr ; then againft 285 
on Bis 183,67 on A, the Perpendicular fought. 

To find BC. 
Set 32^ 48' on Ss to Radius 90^ on Sr, then a- 
gainft 183,67 on B is 339,06 on A, which is the 
Hypothenufe required. 

5. Cafe II. 'Given the Angles 8=32^ 48', and 
C=5s57° 12^ and the Side AC=i83,67 •, to 
find AB, and BC. 

The Bafe made Radius. Fig. II. 

As the Tangent of the Ang. 8=32^ 48'=9.8o9 195 
is to the Side .... AC=i83,67t=2.26403g 
So is Radius .... 90^ 00^=10.000000 

to the Side or Bafe . . . AB==2 85=2 .454845 

To find BC. 

As Radius .... 90° 00'= 10. 

is to the Side .... AB=285= 2.454845 

Sois theSecantof theAngleB=:32^ 48^=10.075428 

to the Side or Hypoth. BC=:339,o6=s 2.530273 

NoUj I have wrought this laft in Secants for Variety 
fake, and that the Reader may fee the Conclu-* 
fions are the fame every Way. But this Cafe is 
much better refolved by making BC Radius, as 
in Fig. I. 

6. By the Plane Scale and Compajfes. 

As there is no ^Line of Artificial Secants on th^ 
Scale or Sliding-Rule (as being ufelcfs) fo this Caf? 
will be beft performed Inftrumentally by the Analogies 
of Fig. I. where BCis made Radius. 

Thcrqfo.«^ 



1 76 Solution (f Plain Tr i an gl e s J 

Therefore, extend the Compalfes from 32^ 48' to 
57^ 12' in the Line of Sines, the fame Extent will 
reach from 183,76 to 285 in the Line of Numbers 5 
thus ^85=AB, the Side required. 

Oi: thus. By the firft Analogy of this Cafe of 
Fig. II. extend the Compafles from 32^ 48' to Ra- 
dius 45 in the Line of Tangents, the fame will reach 
from 183,67 to 285 =AB (as before; in the Line of 
Numbers. 

To find BC. 
Extend the Compaffes from 32^ 48' to Radius 90 
in the Line of Sines, the fame will reach from 183,67 
to 339,o6=BCi in the Line of Numbers. 

7. By the Sliding^Ruk. 

To find AB. 
Set 32^ 48' on Ts to Radius 45^ on Tr 5 then a- 
g^ft 183,67 on B is 285s=AB, on A. 

To find BC. 

Set 32^ 48' on Ss to Radius 90^ on Sr ; then a-, 
gainft 183,67 on B is 339,06=60, on A. 

8, "Cafe in. Given the Angles B=32° 48' and C=: 

Sj^ 12', andtheHypothenufeBC=339,96i 
to find the Sides AB and AC. 

The Hypothemife made Radius. Fig. I. 

As Radius .... 90° 0(y.=io. 

is to the Hypothenufe BCc=339,o6=2. 530275 
So is the Sine of the Angle €=57^ i2'=9.924572 

' 'J ' ' ' 
To the Side or Bafe - . . . AB=:2 85=2.454845 

And fois theSine of the Ang. B=32° 4-^'^9'7 337^5 

. to the Side . . . • AC=i 83,67=2^264038 

Or 



chap. XI. Solution of Plain Tr i an ol es. 177 

Or thus; AB made Radius, Fig. II. To find AC 

As Radius .... ' 90^ 00=10. 

is to the Side .... AB=:2 8 5=2 .454845 

So is the Tangent of ... . 3=32° 48^=9.809193 

to the Side .... AC=fi 83,67:^=2.264038 

9. - By the Scale and Compaffes. 

Extend the Compafles frcwii Radius 90 to 57^ 12' 
in the Line of Sines, the fame Extent will reach from 
339,06 to 285=AB, in the Line of Numbers. 

And then again, extend them from 90 to 32^48^ 
in the Line of Sines, the fame will reach from 339,06 
to i83,67=AC, in the Line of Numbers. 

10. By the Sliding-Rule. 

This Cafe may be folved by once fetting the Rule^ 
thus ; Make the Line of Sines to Aide by the Line 
of Numbers: Then fct Radius 90 on Ss to 339,06 
on A J thus aeainft S7^ i^' ^^ Ss you/ee285=AB, 
andagainft 32^ 48' is;i83,67=AC, on A. Such is 
the great Conveniency of this fmall Inftrument. 

11. Cafe IV. Given the two Sides, AB=285 and 

AC=i 83,67; to find the SideBC, and the 
Angles B and C, 9 

The Bafe made Radius, Fig. 11. To find B, 

As the Side .... AB=2 8 5=^2 .454845 

is to Radius lO- 

So is the Side AC=ii83,67=2.264038 

To th^ Tangent of the AngleB=32^ 48'=9-8o9i93 

Qx thus, making AC Radius, Fig. IH. To find C. 
A a As 



178 Sofutm^ Plain TmAiiQhBs. 

Asthc Side AC=;i 8 3*67=2,2 64038 

is to Radius lo. 

So is the Side AB=:2 8 5=2 454845 

' ' t 

To the Twig, of the Angle C=57^48'=io.i^o7Q7 

12. To find the Side BC, 

This may be done direftly with the Secants, or by 
finding the Angles firft, with the Sines ; as is mani^ 
feft mm the Synopiis. But fince tf the required Side 
BC be nude Radius^ it can't be found by the com- 
mon Analogies of Trigonomelrj ; 1 have given an 
i^quattoii torjoed on t he Principles of Geometry for 
that putpofe, viz: VAfiq-f-ACqasBC i, which I 
ihall refolv^ by Logarithn^ as follows. 

The Log. of the Side . f . AB=:2 85=2 .454845 
the Pouble thereof) .t, o,^^ ^ 

is the Log. of p ' ' ABq=8i225=4.90969Q 

Akb the Log. of the Side AC=:i 83^67=2.264038 

Add the Square • , , . * ABq=8i225 
'"^^thJS*^} ABq+AC(|=ii4?59.65=5.o60546 

L» .1 ,' I. I I I 

Hal f which is the Log. of 

i^ABq4rACq=BC=:j39,o6=s2,5jo273 

the Side required, ^ - 

1 3. By Scale and Com^ajfes. To find the Ang. B. 

Extend the Cdmpalfcs from 285 to 183,67 in the 
Line of NiMnbers^ the fame will reach from Radius 
45^ to 32^ 48', inthe LineofTahgents, th0 Angle 
B required* • 

By 



chap. XI. SfiktfOft ^ Plain Tft i A N GL & s. 179 

# 

By the $lidiHg'Rule. 

Set 285 on A to 183,67 on B, then againft Radius 
45^ on Tti IS 32° 48'=: the Angle B, on Ts. 

Having thud found the Angles^ the Side BC 19 
found as in the foregoing Cafes. 

14. Cafe V. Given the Hypothenufe BC=:339,g6 
and the Side ABe=:285 ; to find the Angles 
B and C, and the Side AC. 

To find the Angle C* Fig. I. BC Radius. • 

As the Side * . * * * BC=3 39,05=2.530275 

is to Radius ' 10. 

So is the Side .... 4 ABral 8 5=2 .454845 

To the Sine of the Angle Csezgy^ 1 2'sr:9.924572 

To find the Side AC. 

As Radius ...««,« to, 

is to the Side ... * BC=a233Q,o6=2.53027-3 

So is the Sine of the Angle Bsas32^ 4^ '==9733765 

fcl II- ■ ■■■■Mil ■> 

to the Side ./.*.* AC:2= 183,67=2*264038 

Nvfe^ The Side AC may be foupdGeotneMcaify^ 
as taught in A rt. 12* the Equation being 
•BCq--ABq=:AC. 

tg. By Scale and Compajfes. To find the Angle C. 

Extend the CompafTes from 339,06 to 285 in the 
Line of Nunibers, the fame will reach from Radius 
90^ to $y^ 12'BaC, in the Line of Sines. 

By the Sliding' Ruk. 

Set 339,06 on A to 285 on B, thep agalnfl Radius 
90^ on Sr is 57^ 12WC, on Ss. 

A a z The 



% 



1 8o Solution of Plain T R i an gl es. 

The Angles being thus found, the Side Xc may 
be found by Scale or Sliding- Rule ^ as before. 

i6. Cafe VI. Given the Side AC=i83,67 and BC 
s=s339,o6 ; to find the Angles B, C, and the 
Side AB. 

As this Cafe is, in the Nature of it, the fame as 
the laft, fo the Solution is in all refpe&s the fame, and 
needs not be repeated. 

ij. Of Oblique-angled Plain triangles. 

Cafe L There is given the Angles C=82^ 30', 
B=6o^ 00' ; and the Side AB=365 ; to 
find the other two Sides AC and BC, 

Note^ The firft Triangle in the Synopfis of Oblique 
Plain Triangles is that which I have regard to 
here, and is acute-angled ; the 'D'lSttttict be- 
tween this and the obtufe-angled one, Fig, II. 
will be taken notice of as I go along. 

To find the Side AC. Fig. I. Co. Ar . 

As the Sine of the Angle C=82^ 3o'=o.oo3732 

is to the Side AB=:265°=2. 562293 

So is the Sine of the Angle 8=60^ oo'=i:9.93753 1 

to the Side fought .... AC=3 18,82=2.503556 

To find the Side BC. Co. Ar. 

As the Sine of the Angle Ci=82° 30^=0,003732 

is to the Side AB=365=2.562293 

SoistheSineof the Angle A=37^ 30^=9.784447 

to the Side required . , . •BC=224, 11=2.350472 

18, 



chap. XI. Solution of Plain Tr i an g l e s. i 8 i 

18. By the Scale and Compajfes. To find AC. 

Extend the Compafles from 82° 30' to 60^ in the 
Line of Sines, and the fame Extent will reach from 
365 to 318,82 in the Line of Numbers 5 therefore 
3i8,82=AC. 

To find BC. 

Extend the Compafles from 82® 30' to 37^ 30' in 
the Line of Sines, the fame will reach from {6^ to 
224, 1 i=6C, in the Line of Numbers. 

19. V By the Sliding-Rule. 

To find AC. 
Set 82^ 30' on Sr to 60^ 00' on Ss ; then agafnfi: 
365 on A, you have 3 1 8,82= AC, on B. 

To find BC. 

Set 82'' 30' on Sr to 37° 30' on Ss ; then againfl: 
365 on A, you have 224,1 i=BC, on B. 

But fince alHhe Angles are known, both the un- 
known Sides are found at once fetting the {lule thus ; 

Let the Line oi Numbers Aide by the Une of Sines ; 
and fet 182^.30' on Sr to 365 on B, then againft 
60^00' is 3i8,82=AC ; and againft 37^30^ is 
224, 1 r=BC, on the Line of Numbers. 

20. Cafe n. Given two Sides, AC=3i8,82 and 

BC=s2 24,11; arid the oppofite Angle A= 
37° 30' •, to find the Angle B, and the other 
Side AB. 

To find the Angle B- Co. Ar. 

. As the Side ...... BC=224,ii=.7.649528 

is to the Sine of the Angle A=37^ 3o'=9.784447 
So is the Side AC=3 18,82=2.503556 

50 the Sine pf the Angle . . B=;6o^ Qofz^^.g^jSi^ 

The 



l8i * Solution of Plain ThiAngIeS. 

The Angles being thus known, the Side A6 U 
found as were the Sides AC and BC in the foregoing 
Cafe. 

21. By the Scale and Compt^es, 

To find the Angle B. 

Extend the Compares from 224,11 to jtS,S2 in 
the Line of Numbers, the fame Extent will reach 
from 37^ 30' to 6qP oo's=?B, on the Line of Sines. 

By the Sh'ding-Rulc. 

Set 224,11 on A to 318,82 on B, then agamft 
37^ 30' on Sr is 60° 00' acB, the Angle fought, 
t>\\ Ss. 

Noie^ In this Cafe of the Obtufe-angled Triangle, 
Fig, II. 'tis obvious the Angle here found is the 
outward Angle CBG,whofe Complement there* 
fore to 180 Degrees is equal to the internal ob*' 
tufe Angle ABC=i2o^ oo^ 

22. Cafe III. Given two Sides AC » 318,82 and 

BC 224,11 ; and the included Angle C=* 
82^ 30^ i to find the Angles A, B, and the 
Side AB. 



To find the- Angles A and B. 

The Sum of the given Sides is AC+BC=tt542,93 ; 
their Difference is AC — ^BCsx:94,7i •, the Sum of 
the unknown Angles A4'Ba«97^ 30' j therefore 

the half Sum is ^ ^ 48^ 45' S whence by the A- 

nalogy in the Sypopfi^, .6y 5 . . . . 



Ai 



Chap. XI. SoJution of PlainTjtiAKQLES. 183 

*".£'SM« "'}• • • AC+BC=54..93-.7-'«i5^^6 
is to their Difference AC-^BCrs 94,71=1.976396 
So is the Tangent of 7 . ^ 

the half Sum of the J ^ =48^ 45^=10.05701 2 

unknown Angles 3 

Then to the half Sum of the Angles 48° 45' 
add the half IXfference , . . . 1 1° 15' 

The Sum is the greater Angle B=6o° 00' 

But the PiflF. is the leffer Ang. A=3f 30' 

?5. By Stale and Compafes, 

To find the Half PifFerence of the Angles ^— ?, 

2 

Having prepared the Work «fi above, pcoceed thus s 

Extend the Compares from the Sum of the Si^es 
542,93 to their Difference 04^71 <m the Line of 
lumbers ; *with this Extentifet one Foot of theCom- 
pafies in Raditxs 45^, and pitch the other downwards 
in the Line of Tangents, where fix it while you faring 
the other Foot from 45^ to 48° 45^5 then with this 
Extent apply one Foot in 45^, the other will tieach 
to 1 1^ ly =5 thcL half Ji^ifference of the Angles A and 
jB ; wl^ich dierefore may fae found as befbr^. 

Or if vou have a Line of Tangents coistiiiued ber 
yond 45^9 th)Mi the Extent from 94,71 to 542,93 in 
^the Line, of Numbers will reach from 48^ 45' to 
f 1^ 15' in the Tangent Lit\e. 



184 Solution (f Plain Tri angItEs, 

24. By the SUding-Rule.^ 

Stt 54-2.03 on A to 94»7i on B ; then againft 45^ 
on Tr obferve the Degree and Minute on Ts, and 
bring that Point to 48° 45' on Tr i then againft 45*^ 
on Tr you have 11° 15' on Ts, which is the half , 
Difference as before, of the two enquired Angles 
A and B. 

Or thus, if the Line of Tangents be continued on 
the Slider beyond 45° ; fet 542,93 on A to 48^ 45' 
pn Ts, then againft 94,71 on A is 1 1^ 15' on Ts. 

Having therefore the Half Sum^ and Half Diffe- 
rence of thofe Angles, they are found as in Art. 22. 
and then the Side AB will be found to be 2^5 as in 
Cafe I. hereof, 

25. Cafe IV. Given all three Sides AB=365, AC= 

318,82, and CB=224,ii, to find the 
Angles. 

In order to this 'twill be neccflary to reduce the 
©blique Triangle into two Right-angled ones ADC, 
and BDC, thus ; Find the Sum of any two Sides 
AC4-CB=»542,93 •, and their Difference AC — CB 
5=::94,7i 5 efteeming the other Side AB the Bafe, the 
Difference of whofe Segments AD — DBsasAG, is 
firft of all to be found by this Proportion, viz^ 

Co. AjT. 

As the Bafe AB:^365=.7.437707 

is to the Sum of I * ^ . o^ ' ' ^ ^ 
the two Sides J AC+BC=542,93'=2.734744 

So the DifF. of the J .^ nr^ . ^, . «^^o«<: 
two Sides I AC^BC«94>7i=^.976396 



J- . 

Thereforo 




Chzp.XL Solutim of Plain Tkiasoles. 1^5 

Therefore AB— AG=224=GB=2BD ; there- 
fore 5H==:ii2i and AD=253 ; and fo the whole 
-T^^gleACB is reduced to two Right-angled ones 
Apfc ancJ*BDC», in each of which there is two Sides 
IJSlD and.4X^ DB and BC 5 by which means 
jles ACD^'and'DCB may be found, which 
_ 3er arfe iB^ual to" the Angle ACB ; and this be- 
^Itnoyn^-Ullye other two are found with Eafcby 
iJaft^ Andtliusth^ whole oblique Triangle isre- 
fiailv^d. 

l^ie^ When the Perpendicular falls without, as in the 
obtufe-angled Triangle ABC, Fig. II. then \t 
will be thQ Sum of the Segnlents AD+DB— 
AG ; and the Difference of the Angles aCD— 
BCD= ACB, the Angle required. 

26. And fince the feveral Problems ofNavigaticn^ 
whether in the Plain^ Mercator*s^ Middle Latitude^ 
Oblique^ or Traverfe Sailing ; as alf^ of meafuring 
Heights and 'Diftancesj accejjible and ^inacceffihle ; of 
Fortification^ Gunnery ^ and divers F^rtf.of ^ronon^^ 
&c. are all refolved by the Doifrine of Plain Trigo- 
nometryj as in the Method before-going ; it muft be 
very eafy for any who underftands the Solution of 
Pkin 'Triangles, to apply it to any praftical Cafes 
that may occur in any of the aforefaid Arts, wirt- 
out any farther Inftruftions or Examples. Yet thofe 
who would fee the Theoty of Plain Trigonometry, and 
its application J in thelargeft Extent, may fitid it in 
the firft Vol. of my Toung Trigonometer*s Guide,, 



B b C H A K 



i86 SobtmdfSphfricalT^tAnoLs^ 

CHAP. XII. 

5i&^ Solution (jTSPHERicALTiliANetES fy 
Logarithms, Ay Cunter'Jt ^cAle^ dm 
by the Slidino Rule. 

1. A S in the fbregoing Chapter I have exempli* 
Xx fied the Refolutjon of Tlain '^ianl^s bbl^ 

hy tht Canon of Logarithms^ and l^arilhmc'al In- 
Jirumcnts^ fo I fliall puriue the lame Med)od here 
with refpcft to Spberkal Triangles of b6tti kinds \ 
in each of which there are Jffx diff'erent X^afes^ which 
in all their Variefiis are refolvable fording to the 
jinabgies ^llignred in the Synapfis % o^ which^ and on 
the Figure of the Triangle thetr, the Reader is dc- 
fired, to have his eye, thro* the who|c Courfe of Ex? 
ampler. To begin then wjth . 

2 . ^ kigbt-angled Spherical TridngteSi 

Cafe I. Given the Bafe Al^=38*^ 15^, and Ahgfeiit 
BafeB==39^ 5& to find the Perpendictihr 
AC, the |lypothenufefiC, and the AnglpC. 

f . To find thp Pcrpendioiikr AC. 

As Radius ..... 90^ oo'=i 0.00000a 

is totheSineoftheSide ABzzjiS^ i^'= 9^791757 
So is the Tang, of the Ang. 8=39^ s^'^9:9'^'^7^7 

to the Tang, of the Perp, AC=27^ 23'j=9.7i^54^ 



d. To find the HypQthfanufc BC. 

As Radios ....... 90° oo'=i6.ooooo6. 

to the Co-Sine of the Angle 6=39° s&zzL^M^G'jy 
$0 i$ t^tOa-T. of thcSidf AB=38<5 i5'=:io.i032jB8 

to theCo-T. of the Hypoth. 80=45^ 48'=:9.988o65 

3. To find the Angle G. 

A$ JRgdjiu? . . ^ . . . . . 90^ oo'=:i6.ooooo6 

t;Q 4c<^<p-Smfi of .the Side ABrsjS^ 1 5'=9.895p45 

., §9 13 .the Sipe of , the Angle 8=^9^ 56'=^. 807465 

to the Co^Sine of the Angle €=59® 44^=9.702510 
3. '^jr ^^7/ and Compaffesi 

I. To find AC. 

. Extend the Co^npafles froni ^o^ to 39^ $& in the 
Line of Sines, that will reach from 38"^ 15^ in the 
jLioci^of J^ngeiits ttp 27^ 23'j=AQ the Side re- 
J5[uircd. 

2. To find the Hypoth. BC. 

Extend the Compafies frooi Radius 90^ to the 
^Po-Siie. ,Qf ^B, 50^ 04' in the Line of -Sines •, then 
.^pply th^t Extent from 45^ in the Line of Tangepts 
down Wards, ^hc^eiix that lower Foot, and bring 
4he.other-to 51^45' the Co- Tangent of AB^ this 
Jaft Extent. will reach from 45^ to 44^ 11' thc,Co^ 
Tangent of BC=45^ 4^'* ^^^ ^^^ required. 

3. To find the Angle C. 

Extend the CompatTes from 90^ to 51^ 45' the 
Co- Sine of AB, the fame will reach from the ^ino 
^39"^ 5^' to 30^ i& the Co-Sine of 0=59^ 4V. 

Bb 2 4* 



i88 Solution of Spherical Tki ah gles. 

4- By the SUding^RuU. 

1. To find AC. 

Set 90^ on Sr to 38® 15' on Ss; then againft 
39° 56' on Tr i$ 27° 23'J=AC, on Ts. 

2. To find BC. 

Set 90 on Sr to 50° 04' the Co-Sine of B, on Ss ; 
and mark the Degree and Minute on Ts againft 4^^ 
onTR, bringthat Point to 51^45' (the Co-Tangent 
of AB) onTR, on which againft 45^, you have 
44^ 12' on Ts, the Co-Tangent of BC==45^ 48^, as 
required. 

3. To find the Angle C. 

Set 90^ on Sr to the Co-Sine of AB, 51° 45' on 
Ss ; then againft 39^ 56 on Sr is 30^ 16' on Ss, the 
Co-Sine ofC=:59^44', as required. 

5 . Cafe II. (ji ven the Perpendicular AC=2 7^ 2 3', and 
the oppofite Angle B=39° s^' » ^o fi^d the 
Side AB, the Hy pothcnufe Bp, and Angle C. 

TofiidAB. 

As the Tangent of 6=39^ 56^=9.922787 

IS to the Tangent of • • AC=27^ 23'i=97i4544 
So is Radius .... 90^ 00^=10. 

■'°li^e%f=?'} AB=38».5'=979.757 

The Operations of this Cafe being only thcConverfe 
of the foregoing, needs no further Examples, iri 

Numbers. 

6. 



ChXll. SpktiM of SpbericaiTKi AUQL^s. 189 

6. By Scale and Compares. 

1. To find AB. 

Extend the Compaflcs. from 39^56' to 27^ 23^1 
on the Tangents, the fame will reach from 90^ on 
the Sines, to 38^ i5'=:AB. 

2. To find BC 

Extend the Compoffcs froifa 39^ $& to 27^ 2^% 
on the Sines, the fame will reach from 90!^ to. (the 
Sine of BC=). 45^ 48^, as required. * 

* 3. To find the Angle C 

Extend the CompafTes froni (the Co-Sine of AC) 

62^ 36^1 ^d 90^, the fame will reach from. (the Co- 
Sine of B) 50^ 04' to 59^ 44' the Sine of the Angle 
C required. 

7. ' By the Sliding' Rule. 

I. To find AB. 

Set 39° s^' onTR to 27^ 23'^ onTs, then againft 
90^ on Sr is 38^ i5^=AB, on Ss. 

2. To find RC. 

Set 39^ s^' on Sr to 27^ 23^1 on Ss 5 then againft 
90°.on Sr is 45^ 48'=BC, on Ss. ' 

3. To find the Angle C. 

Set (the Go-Sine of AC) 62® 36'j on Sr to 90^ on 
Ss ; then againft (the Co-Sinc of B) 50^ 04' oh Sr is 
59^ 44'=^C, on Ss. 

8. Cafe III. Given the Hypothenufe BC=r450 48', 

and an. acute Angle B=39*' c&^ to find the 
Legs AB, AC, and the Angle C. 

I. 



1^ $M«ii ^Spt&icalfniAtmiisi 

i. To find AB. 

As kadius .<*,.. .' . 90° oo'sasio.oooooc? 
to the Co-Sine of . . . lis=39° 56'=?; ^M4J^yf 
So is the Tangent of . < BCas^^° ^8'=io,oj2?29 

As Radius ..*.••* ^d^ 6(5(is=ip, 

is to the Sine of , ^ . ; ''BC*:4:5<^ 48'-? 9.8/54^^' 

Sois theSin6 of • • « . B=;=39^ 56'=s9.8o7465 

fought - } AC»;=17^ 2fjt^^Mm^ 

3. To find the Angle C. . 

As the Co-Sinie of . . . .BC;=i^^ 48'=9.84333_<5 
is to Radius . • . . ^ 90 00=10. 

So is the Co-Tang€nt ofBii!i339^ 5&s=zio.oyy2iif 

■■ - '1 

Toithelfaiigentat. . . ^C;azg^^ 44^10.^3387/ 
9. By Scalfi ,and Qifnpajes. 

^. XocfindAB. 

Extend the Compflfes iwm Radius .9oP4»^(^ O^ 
fine of B) 50^ 04' in the Sines, then fet one Foot iii 
45° in the Tangents, arid pitdhthe other downward ;, 
^here &c it^ while you bring the former from 45^ 
to 45^48^4 then will this laft Extent reach from^j;^ 
to the Tangent of 38^ I5'=3 5lB^ theSidc peqqircd. ' 

2. To find AC. 

^Extend the Compafles from 90^ to 45^ 48' in the 

Sine$» 



Sines, the fame will reach frofp. sg° s^' IP ??** '^3fi 
SsAC, in the^kme Line. 

3. To find the ^leC. 

Extend the CompstfTe; from (the Qo-Sme of BC) 
44° 16' to Radius ^^ j the fame applied to (the G>- 
T«agent of 8)50^94', wiJl t«acji>»fh^ Tfsgent 
5j>^ 44'*T^» the Angjie Mqujied. 

I. T<> iiod A?. 

Set Radius 9p° on Sj| to 50° 04'^ CVSbe^f B) 
ion Ss, and a^inft 45° on Tr mark the point oa T«, 
"bring that Point to 45° 48' on Ts. ;j and now agaiaf^ 
45° on 



hat romt to 45^ 48' on Ts. ^ and now aeunll; 
-° — Tr you hav§38° i5'=AB, tm Ts, 



?.ToiSndAO. 

Set R^dius^o" on S to 45* 48' on Sr, then ^gaioft 
39* 56' on Se, ypu'Jl fee 27" 23'i on Sr, £hc Side 
AC eegtiire^. 

3, To find the Angle C. 

Set (the Go-Sine of BC) 44* i6'on Ss to Radius 
go* on Sr j then againft (the Co-Tan^nt of fi) 
Sp'oVonTR, js59<'44'=;C, onTs. ' 

II. CafeJV. Given the Xep, ABz^jS' 15', an4 
AC— 27' 23'j } to W the reft: 

To find the Hypothenufe BC 

As Radius ....,,. 90° oo'asio, 

^s to the Co-Sineof . . AC=a:27''23J=9.948388 

So is the Co-Sme of . . AB=38* i5'w:9.895045 

•fo^he Go'Sine pf . , . , ^Csss^is" 48'ss?9.843433' 



19^ Solution of Spherical Triangles; 

As there is nothing new in finding the Analogies B 
and Q I fhall pais them by. 

12. By Scale and Compajfes^ 

I. To find BC. 

Extend the Compafles fi-om go"" to (the Co-Sine of 
AC) 62* 37'; the fame applied to (the Co-Sine of 
AB) 51^45' will extend to (the Co-Sine of BC) 
44'' 16' ; therefore BC=45^ 48'. 

2. To find the Angle B. 

^ Extend the Compafles from 38** 15' to 90^ in the 
Sines, the fame will reach from 27"* 23^1 to 39*^ 56' 
=8, in the Line of Tangents, 

3. To find the Angle C. 

. Extend the Compafles from 27° 23^1 to 90** in the 
Sines ; the fame will reach from 45'' to a Point in the 
Tangent- Line, where fix the Compaflb while you 
bring the Leg from 45^ to 38'' 15', then will this laft 
Extent reach from 4^° to 59^ 44' =C in the Tan- 
gents. 

13. By the Sliding' Rule, 

I. To find BC, 

Set 90'' on Ss to (the Co- Sine of AC) 62** 36'! on 
Sr, then againft (the Co-Sine of AB) 51"* 45' on Ss 
is (the Cp-Sine of BC) 44"" 16' on Sr •, wherefore 
BC=r45''48', is required. 

2. To find the Angle B. 

Set 38*" 15' on Ss to 90° on Sr ; then againft 
27"" 23^1 on Tfs is 39"" 56' on Tr, the Angle B re- 
quired. 

3 



Ch; Xlii Siihaiwi ^-^y^^itikviGiis. i^ 

0; To $fid the Angle is. . 

^ 27^ 23' ofi Ss to s^ b'h Sit ; mi. fiistrk the 
Point on Ts agairift 45^or< X'» i bring that Point tO* 
38® <5' on Tr J, thc&agaihflr 45* ofi T* Joi ft« 

55^? 44'^c;'6tvT«.- ' •••..■■•••. 

i4. afeV. Gf7etitheHyp6thcinjfeScd?4^'»48'. 

and one $i4c ^Cata?** 2.g,'4 ; to fiQ<^^(^,rC^ 

Tb gnd the Angli; C. 



r t 



As the Tangent of A^&tj^ ^J^i^is g^ 14.5^44 
Is to the do-SlAe of the hti^&i^^sQf^4/^'^^.jCii/^i$ 

ii . : ^jf^t^.s£^et^dJ^ V;."': 

.' Extend fliA^dotni^ffesfrom45^to«5^f?i^fy^ 
Taiigcitisy irhen fet bne foot in 4^^ 48.^, and pitcK 

the other again frOni 4,^^.4^' to 45^: thcfi (haff thrif. 
laft extent reach torn HacHtts jro^ w(the Co-Sine'of 
C) 30^ i& in thfe Line of Slices 1 wheitce the Angfe 
Cs=59° 44^, as reijuired. 

i& iStyi/ie Slidhjl'kule;' 

$6t4^6xi Tr^p 27"^ ^V^5t'^^^! ahd^afk thi 
fbirit inT^'ag^ntf 45^4S'*dA Tir, then brjrijf tha^ 
Point to 45^ on Tr ; laftly; againft-9C^oh»*; yoii 
have 30^ 16' ott Ss^ w^ic^ is, theCo-Sfne of C*±y 
59^44^^ asrequrrett: * ''",';' 

'ATfl/i, TJh^lProportioWi M A*haMtfc:AngJc;1ft, 
contain! hothtngtitw. or diflScube^r ib \Niwmrs 6i 
by Inftirument> thecc^o!re.flslltrgiTe:noi £xao!ipIes' icf 
them. ' .. • 



1 



t;. CaTe VI. Given the Angles 63=39° 56', and 
Crs59° 44', to find the Sides AB, AC, BC. 

I. To find AB. 

As the Sine of ... ; B=39® 56'=:^.8oy4.6s 

is to the Co-Sine of . . . ^=59*^ 44'=9>702452 

• JSnLa Radnia acP Ort'r— Tr» 



So is Radius 90^ oo'=sio. 



To the Co-Sine of the ;Side ABss^Z^ «5— 9.894987 

2. To find AC. 

The Analogy is the fame as for AB, by which yoo 
ynjl find AC=27^ 23^1 nearly. 

^ 3- To find BC 

As the Tangpnt of . . . €=59^ 44'=ip.2 33905 
ifi to the Go-Tangeht of Bss^gl^ s&^io.oyyu^ 
So is Radius • . . • . 90^ 00^=10. 



to the Co-Sine of the Side BC=45° 48'=. 9.843308 

1 8. By Scale and Con^ajjfes. 

1. To find AB. 

Extend the Compaffes from 39^56' to (the Co- 
Sine of C) 30^ i6\ the fame will reach from Radius 
00^ to (the Co-Sine of AB) 51^ 45', in the Line of 
Sines -, therefore AB=38^ 15'. In the fame mannei; 
you find ACssz;^ 23'!. 

2. To find BC, 

"T^xtend the Cot^pafles from 59^44' to (the Co- 
Tangent of B) 50*" oV in the Tangents, the famo 
Extent will reach from Radius 90^ to ,the Co-Sine 
of BC) 44^ 12' in the Smesj lhd-cforcBC=45^ 48'. ; 

^9' 



19. By the SKdif^-Ruh. 

1. To find AB. 

Set 39® s& on Ss to (thcCo-Sinc of Q 30' i & o^ 
Sr } then againft .90^ on Ss Is (the Co-Sine of AB) 
51^ 45' ; confcquently AB=38^ 15', 

In like manner, you find ACsssa;^ 23'!, 

2. To find BC 

Set M^ 44^ on Ts to (the Co-Tangent of B) 
50^ 04' on Tr I then againft 90* on Sr is 44** 12' 
(theCo-SineofBC)onSs. Wherefore 80=45^48^ ^ 

2a. Of OMique Triangles. 

Oblique Spherical Tmngles admit alfo of fix difr 
ferentC^fes, as follow. 

Cafe I. Given two Angles, 8=34® 30^, and Dsa 
48'' oo^ and an oppofite Side BCS38'' 45' ; 
to find the other two Sides DC and BP ^ and 
Angle Q. 

Let fall a Perpendicular CA from -the unknown 
Angle C to its oppofite Side BD. Then is the ob- 
lique Triangle reduced to two Right ones, BAC tod 
D AC. Then (ay by the firft Analogy 5 To find the 

Angle C 

As the Co-Sine of . ... BCsr38^ 4«'s:9.892030 
^ 13 to Radius . . • • • gQ° op^ip. 

So is the Co-Tangent of 62:34'' 30^=10.1 62 &6< 

to the Tangent of . . BCA^ei"" 48'f=io.27a?36 

Again, by the fecond Analc^y, &y } 

Cc 2 / A4 



1^ S(i^ipif'ffpkfr4^ 

Co. Ar. 

AsthcCo-Siiicff , . . Bs??*"* 3p'=p.o84oo^ 
is to the Sine of. . . )BCA==6i^ 48'l— 9-945159 
So h the Co-Sine pf . . .' 0=48"* po'=9;8255i i 

fptJieSineafthp %fi^t If C Az=:^'' j,i^z=^. 3^4676 

l^o?jr,finiCie the Veipendliarl^V CA felk ^vithih th© 
Tnangle', ^tisplaip tKeSum of the two Angles noi^ 
found makes the Apgle.fopght^ viz. BGAj4-DCA=: 

, ' '• ' ' ^ 

To find tl» Side BD^ 



SI. By the irft Analogy, fay ; 

As Radius ...... . 50° oo'==ip. 

to th« Co:Sin? of 5=34° 3° -9-9 ' 5994 

So is the Tji^ht of ' . BC=s:38° 45^=9.904497 

^ -die Tapgenf of the Side BAs^s" zg'=.^.i2048^ 

^i»^« by the Se co n d ^^laloey, % ; 
i " ,' ' ' Co. Ar. 

A? the l*angen> of . . . P=48' 00^= 9-954437 
is to the Tangent of . . . B==34° 3p'=:9.S37i 34 
fipfi the Sineiof . /^ > . AB±:|3? ^9(1=9741698 

to^JPijiepir . ; J . AD=5i^° 57'=9-5332^9 

Now 'ttis Evident, the Sum of the two Arche^ 

AB4.AD=pBD=r33- 29'-t?9° sY.-rSB^^^'* the 
•fetemiulfcd: - • • ' '^^ 

p. ^ . ^o^4|be fye 0p. Co. Ar. 

As the S^e of the Angle p=48'pQ'=:.o.i2892jr 
Js ?o t^'Sineof ^ ^e 6C=338' 45'-=s9.f 9657 1 
5oj?rtejSiffepf jtheAftgle ^==3^ 3o'=9:753i28 

^thc Sins bfthcSide fot^t^C^S" ^o'=^.678626 



Thus tbe whole Triangle is refolved according to 
,»bc Data of this Cafe, .:;;. 

•* ' « 

il3. Cafe 11. Given two Angles Btc34^ gp', and 

C=i=i07° 3P' J and u .Si4e included BG=? 

38^ 45' ; to find the reft. 

You let fell the Perpendicular CA, and find the 
Angle BCAi=6i^ 48'^, y^ jn Art, »2.o, . Thyi, f/ott 
the given Angle Cmic;^ 3p(, take the Angle BCA 
=61^48'!, and there will remain fhe Angle DCA 
v±45^ 41'f, by which you will find the Afigle D;=: 
48^ oo^ according to the fecortd Analogy of this 
Cafe. Tnejre Things being known, we may pro« 
ceecl 

To find DC. Co. An 

Thus, as the Co-Sine of DC A2=45^ 41 '1:^0.155 821 
is to the Co-Sine of BQA:^6 1 ^ 48^1=19.674 :? 29 
So is the Tangeatof , . • BCzrgS^ 45 =19.^04491 

To the T. of the^idc fought DC=28^ 30^=973^6^1 

»4. To find BD. 

From the given Angle B let fall the iPerpendiculat 
BM to the udLflownfSide DC produced } then by the 
firft Analogy, tnd the Angle C^^J, thusj 

As Radius 90^ 00'= I p. 

. to the Co-Sine of ... . BC=38^ 45'=9. 892030 
So is the ttngent of 3BCK=72^ 3q'=:io.50i278 

totheCo Tangept of C9>fc=:22^ Oo'j=jo.39j3o8 

Then by the fecpnd Analogy, fey > 
As the Cb^ine of • . DBh=56^ 3o'j=,©.'2^&2o8 
is to the Co Sinepf NBC3=22° oo'|=9.967i5i 
So is the Tangent of . . . fiC=38°.45'r=9.90449i 

tothcT.pf theSidcfoqghfDC=r53^ 26'^ip:j29%q 



19^ Solution of SpbericalTKijiVOLEsl 

25. Oife III. Given two Sides BCaagS^ 45', and 

CDs=:28^ 30', and an Angle oppofite to one 
of them, B=34^ 30' ; to find the reft. 

This Qfe being but the Convcrfc of Cafe 1. 1 fliall 
not repeat the Examples. 

Cafe IV. Given two Sides BC=38^ 45', and BD= 
53O 26', and the Angle included 8=34^ 30' ; 
to find the reft. 

This Cafe contains nothing difficult if what goes 
before be well underftood ; the Analogies being plain 
'and cafy for the Operations, I (hall leave them to 
the LeamePs Exercife ; and pafs on to 

26. Cafe V. Given all the three Sides, BCsss38^ 45', 

DC=28^ 30', and BDs:53^ 26' ; to find 
the Angles. 

To find the Angle C. 
The Difference of the Legs containing the Angle 

fought, is BC— CD=AM«io^ 15' 5 then ?5±^ 

t^iSi"" 50'h and22z±il=:2i^35'|; wherefore, 

according to the Analogy for this Cafe m the Synop* 
fis, proceed thus: 

The Sine of the Side. . . ^€=^9^ ^s'zsg.ygSs}! 
AddtheSineoftheSide DC=28^ 3o'=9.678663 

The Sum is the Log. of sBCxsDC=i9.475234 

Again, the Sine pfBD+AM ..^ ^a 5o'i=9722283 

2 

; Add the Sine of BD-AM^^^o 35'|=9.565836 



TheSumis sl2+AMxs 22=AM gj, 

z Z ■^ "^ 

Then 



Ch.XII. SolutktrefSpbencalTtiiAHQLii. ipf 

Then iky j 

As 8BCxsDCs=:i5|,475234 

is to the Square of Radius . . . Rqss26. 

„ . BD-f-AM ^, BP— A M „oo,,^ 

So 18 8 --t ** r 8=19.288119 



to the fquare Sine of $ the! c„,r-_»/, o.«bo^ 
Angl2ibught ' }..,SqfC=:i9.8i2885 

The half whereof is .... sfC=:53° 44fs=g.go644t 

Wherefore the Angle fi>u^t is CaBio;^ 28' ; or 
more compendioufly thus, by the Arithmetical Q)m* 
plement oftheSides BC, and DC. 

27. The Sine of BCsrgB® 45'=t0.203429 Co. Ar« 
The Sine of DC3s28° 3o'=.o. 32 1 337 Co. Ar, 
TheSq. of Radius Rq . . s20.oo<X)Oo ... . 

TheSine of £5±AM -3,0 5o'|-9.722283 
The Sine of ^=^ =21° SS'k^B-S^S^S 



The Sum of all . . . . • 8q|C=i9.8 12885 

* 11*1 II 

ThcrcfcMTC . • . . 8|C=s53^ 44^5^9.906442 . 
Confe^uently . . . Cszioy9.ti'i the fame a9l)efi*e. 

Having fbund one Ang)e» the others aife.'eafflf IbUod 
by the former Cafes. 

' jV(7/^.ByCafe I. the Angle C yfas foun^to.be 
107® 30^ which ia but 2' difiercnt frofli what it 19 
found by this Cafe ; whence the Reader may pbierve 
the wonderful Certainty and Agreement of^the moft 
difFereAt Methods of Calcubtiony and fixxoa the grjMt- 
eft Diverfity oiP Data^ ' ^ 

, * : .. .- ^h 

I V 



28. The fixth Cafe Is but the fame with this, tf 
the three Angles giv^n fiecfaatiged into Sides, taking 
for, tl^ gr,eaJteft Aqgh^ in that Triangle and greateiit 
Side in tnisj, their SuppI^ifteptK tQ, iSp.fjegpce^. 
. 29. Having thys Ibiewn the bcft Methods^ both for 
EkaS^filim Eafe, ofMbfving thtf fevehil Cafes of 
kightJxuLOblique'angled Triangles^ I (hall leave the 
Application thereof to ttie.i^noosPnbibkniS'af ^r^ 
i^omy^^'Gedgraphy^'Di tilling^ Ortbvdrsinies^ ^. for 
^he LevPfii^'s own Rc^rtotion 5^ as^^jbg^i? it felfi y^fy 
eafy, if what is here taught 6e uriderftood ; and alfo 
becaufe I have both d^m^tiftrated t^e'T^^^ry, and 
vcryxbfcly. applied i\ic T>9Urint oi Bpberical ^rig^;^ 
nometry^ in the 2d Volt of mj Toun^ ^rigmoiMtef^l 
Guide^ printed for Mr. J. Noon, at the PFbiie HarU 
in VUi^^ { and wB.Ch' Irecommeml^tcr all uhd<r- 
quainted^i^ith, and^nthcLW<9uIdhate a good NotiM of 
the nob]e;dQd moft ofeful Art otTrigmofpeify^^ - 

: , C H^ p. XIIL 
Merca t qj i' ^ Sailimg performed by the Ca- 
non ofho^hioi i-it^H Ktre-T A N gen f^, njottbr 
^f the MsAiDj'air A^ PaAts. / 

l^r«^I*E*^cipferty ^f^Msf^^aidr^s tkdrt is its'hiV- 
J[ ing the Degrees of Lafitwie iWereaicd-^ih the 
fame Proportion as a Pegrce qf JL^n^tpdft decreafes 
friifitfcfe-E^u&ft'rtb the Pole. Whtch' Ptdportibri 
is tha«» of iJfliffiifif to the Co-Sm of Latitude i or, ^of 
the A?rrf*^ of -tfee ^Latitude tor itii Radius '9 which id 
ihus dfcmoiKffirited,^ 

- 2. JLet AEB (F!g;1^ift) be-a Seftor in the Plane 
of the Ecjuator, made by the* Iriterf65lions of the 
l^htxes of two Meridians therewith, viz. 4E, ar ^ 



Cb*Xni.gfLooARiT»Mie Tangents, aj?!. 

BE^ whofc Inclination, or Angte BE A, that is, ^h!^, 
Arch of the Equator AB is = i Degr^* Alio let 
EC be the Radius of any parallel of Latitude ; thca 
iftiall DC be an Arch in that Parallel fimilar to (AB) 
one Degree of Longitude In the Equator, or it is 
that Degree diminiihed. Now (from the Elements) 
the Arch AB is to the fame diminifh'd in the given 
Farallel DC, as the Radius of the Equator AE to 
the Radius of that Parallel CE ; but fmce AE=EC 
(AC bdngthe giVen Latitude) and bstaufe of fimilar 
TrianglesEcCand ESA, therefore Ec (EA) t EC :i 
ES : EA 5: AB : DC. But EC= E c, the Co-Sine 
of the Latitude AC, and ES the Secant thereof y 
therefore, fc?r. which was to he demonftrated* 

3. And becaufe in Mercator^s ProjeSion ot Charts 
the Meridians and Parallels are all reprefented by pa» 
rallel RigM-Lines^ the Arch CD in every P^ralfcl is 
ever equal to that in the Equator AB i therefore that 
the true Proportion between Longitude and Latitude 
might be preieryed on this Chart, as on the Globe it 
felr, 'twas necef&ry the feveral Degrpes of Latitude 
(hould be fuch as belong to thofe Circles whofe Radii 
are feverally equal to the Secants of thofe Latitudes. 

4. Let A =± the Arch of one Degree in any Lati- 
tude ; ^, ^, r, &c. =s: the feveral enlarged Degrees 
on the Meridian ; Raa Radius ; and S, s, j, C5?4 
the Secants of thofe Latitudes. Then it will' 

{Aift : tf :: R : S. 1 Therefore we ftiall have 
A2d :^ :: R:s. igAr^+^+c:: 3R: 84* 

Ajd : c t:K :sy &c. J s-j-i. 

That is, the Sum of the Secants of i, 2, 3 Degrees 
is always equal to theDiftance of the Parallel of thofe 
three enlarged Degrees from theEquatgr. And con- 
fequently by a continual Addition of the Secants of 
f' i|', 2|', 31^ &c. or their Doubles i', 3', 5' 7', 
(^c. the Line or Table of Meridional Parts may be 
made ^ and by fome has been made in this manner 

Dd ^ for 



202 Me RCATOR'i Sailings by the damn , 

for every Minute erf the Quadrant; but a truer way 
is that delivered in Chap. X. of the Theory, which 
fee. 

5. If then you draw any Line AB (Fig. X.) to- 
reprefent an Arch of the Equator, as fuppofc 60 De- 
grees ; and on every tenth Degree be ereded Per- 
pendiculars, thefe fhall be the Meridians ; on which 
if the Degrees of Latitude, enlarged in. .the Manner, 
and Proportion above defcrib'd, be fct ofF; and thofe 
Divifions join'd by Right-Lines, reprefenting the 
Parallels of Latitude, Mercafor^s Chart will be con- 
ftrufted for the given Longitude and Latitude ; and 
a Meridian thus j^raduated, is what is called the 
Nautical or Meridian Line. This Chart, as here 
drawn, includes 60° Longitude^ and 80^ Latitude^ 
that the Reader may have a perfpicuous Idea thereof. 

6. Now fuppofe F be the Latitude whence you 
fail, and C the Latitude you arrive to, on the Rhumb 
FC ; 'tis plain there will be formed a Right-angled 
plain Triangle FCD, wherein FDis the enlar^d Dif'^ 
ference of Latitude^ and CD the true Difference of 
Longitude^ and the Angle CFD the Cc«r/^ or Rhumb ; 
and therefore any two of thefe being given^ the reft 
are found by the Analogies of plain Triangles before- 
going. 

• 7. Since every Degree is equal to 60 Minutes or 
Nautical Miles, let i^^f roper Difference of Latitude 
be reduced to thefe Parts, and fet (off the fame Scale 
as AB was laid down by) from F to E, and draw 
EK parallel to DC ; fo fhall there be form'd another 
Right-angled plain Triangle FEK ; in which FE is 
the proper Diff^ence of Latitude ^ EK tht Departure i 
KF the Dijiance failed on the Rhumb or Courfe EFK. 
Any two of which Parts being given, the others are 
to be found as aforefaid. 

8. And thefe two Triangles FCD, FKE, com- 
prehending all the Particulars of Mercatofs bailing ; 
*tis obvious enough how they are all refolv*d^ either 

by 



Ch.XIII. gjTLoGARn^HMic Tangents. 20 j 

by ProjeSlion or trigonometrical Calculation \ but my 
Purpofe is here to fhew, that "^itkont tho^^ common 
Methods^ they are all to be refolv'd by the Canon of 
Logarithm-Tangents, only -, in order to whicR, the 
following Articles hiuft be premis'd and duly ob- 
ferv*d, viz. ^ ■ ^ 

9. Firft, That it has been (hewn, the Nautical or 
Meridian Line^ or Scale of Mercaior's Cpart^ is no 
other than a Scale of Logarithm-Tangents of the 
Half- Complements of the Latitude. Secondly, that 
fuch Logarithm-Tangents of Mr. BHgg*sForni, (or 
thofe in common Ufej are a Scale of the Differences 
of Longitude upon the Rhumb, which makes an 
Angle of 51° 38' 9^'' with the Meridian. Thirdly, 
the Differences of Longitude on differing Rhumbs, 
are as the Tangents of the Angles of thofe Rhumbs 
with.the Meridians ; as is evident from the Triangle' 
FCD (Fig. X.) And fourthly, that the Logarithm- 
Tangent of the Angle 51° 38' 9^^, viz. 16.10151Q4 
is a conftant Faftor in thefe Computations. On theffe 
Premifes ^tis eafy to operate the Propofitions of Mer- 
cator^s Sailings as follows. 

ID. In Fig. II. letF reprefenttheL/z^j^rJV Pointy 
whofe Latitude North ' DF=;49^ ^p{ ; alfo let K;re- 
prefent Barbadoes in Latitude North 13^ io'=CK ; 
then is FE = the Difference of Latitude ; CD the 
JDifference of Longitude ; and EFK the Angle of 
the Rhumb, FK with the Meridian.. Laftly, let FO 
be the Rhumb making an Angle of 52^ 38' g" with 
the Meridian ; or that on which the Difference of 
Longitude Op is the Difference of the Logarithms 
of the Tangents of half the Complements of the Lati- 
tude PF, PK, or PO: Then, 

II. Cafe I. Given the Latitude of the Lizard^ 

49^ 55' N. and that olBarbado.es 13° 10' N> 

and their Pifference of Longitude 53^ 00'= 

D d 2 3180 




io4 Me ftcAtOR'j SaiHngt iy the Catm 

3180 Nautic Miles r I demand thdCourft 
and nftance M'd ? ^ 

{TheLat. 1 
CK=i3° 10 
DF=49°55 

•Then the Log. Tangent of { f^^^^.^ 

The DifFerence of which 19 . . , QP=3372,6o 

(Note,^ The four firft Figures arc Inte^ers^ according 

to the Theory) 

<■ 

Then fay, for the Courfe KFE ; 
As the DiiT. of Log. Tang. 00=3372,6=3.5279^5 

tothegiveh I>iff. of Long. €0=3180=3.502427 
.So is the cohft.Tang. of OFE=5 1 ^ 38/ 9''^= i o. i o 1 5 1 o 

y ' /. 
3.603937 



to the Tahgent of the 7 rrr-T? o / 

Courfe f?ught |KFE=49^ 59^1=1 0.0759 7^ 

Secondly, For the DiftanceFK. 

The Difference of Latitude is 49^ 55' — 13° io'=* 
36^ 45^=2205 Miles ; 

Say, as Radius 10. 

to the Diff". of Lat* • .. • !EF=2 205=3. 3434091 

So is the Secant 1 vt:>t> o / // o 

oftheCourfe { KFE=49^ 59' 10^^=10.191807 

to the Diftance faii'd FKi=:3429,38=3,5352i6; 

By this Propofition you eftrmate the Courfe a Ship 
muft fteer, and the Diftance of her Port. 

12, 



Ch.XUI. ^Logarithmic Tangents. 205 

12. Cafe 11. Given the Latitude of the Litard 

49^ 55' N. zxiAo( Barbadoes if 10' N. and 
fhe Courfe 49^ 59^, to find the DifFct^nce of 
Lx)ngitude, and Diftance failed. 

Things being prepared as before, fay j 
As the Log; Tang, of OFE=5i^ 38' 9^'- 10. 101 510 

•".S'cwf- °'} 1^^=49° «' .o'«.o.o7597. 

So is the DifF. of the Log. I 

Tang, of the | Cpmp. >Qp=s*3372,6^3.5ft79€5 
of the Latitudes J - . . :^ 

3.603937 

wir ifi'r^ 

to theDiff. ofJl40ng. required CDs3B3 1 8osa 3.502427 

Which <x>n verted foto Degrees makes 53^ oq'. 
Andib HHich is the Difference df Motion or Time 
between thefe two Places. The Diftance on the 
Rhumb will be found 3429,3S=z:FK,*as before. 

13. Cafe IIL Given the Latitude of the LiTMrd 

49^^ 55^^' *he Diftance failed 3429,3 8 Miles 
on a Courfe 49^ 59' 10^ South wcfterly, 'tis 
required to find the Latitude and Longitude 
of the Place to which the SJup is arrived. 

As Radius , 10. 

to the Diftance failed FK=3429,38==?3. 535215 
So is the Co^ Sine of J i^ft? o 1 h « « 

the Courfe }KFE=49° 59' io'=9. 808 193 

. to the Diffw of Latitude FE5=52 205=2^3. 343409 

Thfen from the given Lat. of the Lizard 49^ 55' 
Subd.; the Diff. of the Lat. 22053^36^ 45' 

There remains the Lat. {ought . . , =13° lo' 

And 



ao6 Me r c ator V Sailings by the Canon 

. And having thus obtained the Latitudes, the Dif- 
fcrencc of Longitude will be found, ^fer Cafe II. to 
be 3 1.80 Miles or 53 Degrees ; and fo the Operation 
needs not to be repeated. 

14. If it fo happen that the Ship pafles the Equa- 
ior^ an4 confequently has one Latitude Nortby the 
iOther 5^«/^ ; then obferve two things; firft, that 
ioth the Complements of the Latitudes are to be 
eftimated from the fame Pole of the World. And 
therefore, fecondly ; fuppofe you fail from a Norths 
em to a Southern Latitude, you muft add 90^ to the 
fin'tner^ and fuhftrad the latter, from 96^; then fub- 
ftracSt this. Sum and Remainder froib 180°, and take 
the biiFerence of the Logarithm-Tangents of half 
the Remainders, as before. 

15, Cafe IV. Suppofe I failfrom Latitude 48° 30' N. 
. ( to Liatitude'23^ 45' S. on a Courfe 43° 50' 
. Southwefteriy ; required the 'Difference^ Lon- 
i gitude^ and Dijlance failed ? - 

Then 48° 30^4-90=:i28^ 30' ; and 90— 23^ 45'=3 
' 66^ 15^ And 

ig'o— 66^ 15^=113045'!, 556° 52^1=10.185410 
'180—128^30^= 51"^ 30^^125^ ^5' — 9-683356 

The DifF. of thofe Tangents is ... 5020,54 

. Therefore fay, for the Difference of Longitude; 
As the Log. Tangent of 51° 38' 9^^=10.101510 

is to the Diff. of the Log. Tang. 5020,54=3.700750 
* So is the Tang, of the Courfe 43^ 50^=9.982309 

3.683059 



to the Diff. of Long", required 3815,5=3.581549 
That is, inPegrees,=a:63°35^|» 

To 



Ch.XIIL g/'LoGARiTHMic Tangents. 207 

To find the Diftance faiPd 5 fay, 

.As Radius , lO. 

' to the Sum of the Lat. 72^ ^ 5'==43i5^3^6^6gS^ 
So is the Secant of the Courfe 43^ 50'= 10. 141 849 

to the Dift. faird in Miles, ^oo^^s^zs'.yySS^S 

16. Thus you have all. the PraSical Cafes oi 
Mercatof^s Sailing performed by tYi^Qanon ofLoga--^ 
rkhm-Tangents only ^ without the MeridionaV Parts ot 
Charts as in the common Way. And fince this' is . 
themoft exaft, and natural Method of Navigation 
(next to the Globular Chart it felf) and wholly refol- 
vable by Logarithms^ jt adds not a* little to the (be- 
fore invaluable) Eftimation :of thofe excellent Num- 
bers ; and renders their Ufe to Navigators more ne- 
ceflary than before. 



CHAP. XIV. 

0/ /i&^ Mensuration (j/SuPERFiciES and 
Solid Bodies by Logarithms, 

I. \ Mongft the Variety of Methods for meafuring 
£\ the Surfaces and Solidity of Bodies^ I intend 
here to ftiew the Excellency of that by Logarithms 5 
which may juftly be allowed the Preference to all o- 
thersinpoint of £^y^and JSrm/y, Advantages none 
of the leaft in common Eftimation. And fince thefe 
Operations confift altogether • in Multiplication 3.nd 
Bivijion^ I need not here repeat them by the Injlru- 
ments^ as having already largely fhewn the Man- 
ner thereof in a Chapter particularly on that Head. 

2. 



flo8 Mti^svuATioN of Sn^Jlcses an J 

2. To meafun a S(ii;ARE, Fig. XII. 

The Logarithm of the Side multipKed by 2, ^vcs 
the Logarithm of the Area otfuperficial Content. 

Examp. Let the Side of the Square be AB=c3i,57. 

Then the Logarithm of AB=3 1 ,5731= i .4992 75 
which multiply by 2 

The Produd is thei Aur^rr ^ ^m a ' 

. A«a of the Square {^®^^'^996,66«:2.998550 

3. To meafure a PxtL ALL BhOGK A u^ Fig.XIII. 

Th6 Sum of the Logarithms of the Length and 
Breadth is the Logarithm of the Area. 

Examp. The Length ABs=a4i,5 and-Breadth BC=? 

To the Log.of the Length ABaK4i,5=i.6i8o48 
add the Log. of the Breadth BC=3i,57«=i.499275 

The Sum is the Log. } * wr^TS 

oftheParalIeIog^i^^^^='^3i2,i55=3.ii7323 

4. To meafure a Rhombus ABCD, Fig: XIV. 

The Sum of the Logarithms of a Side and the 
Perpendicular Height, is the Logarithm of the 
Area. 

Examp. Let the Side AB=i 5,5 j and the Perpen- 
dicular BE=i3,42. 

Then to the Log, of the Side AB=i5,5=ri. 190332 
add the Log. of the Perpend.BE^s:! 3,42=1=: i.i 27752 

The Log. of the Area of I ^^n ^^ . ^,0^0, 

the Rhombus i' ==^08,01=2.318084 

5. 



Ch. XIV« Solids iy LoGAiSiiTUM$. 209 

5. Tomeafure a Rhomboides, ABCD, Pig.XV, 

The Sum of the Logarithm of the longeft Side^ 
and perpendicular Height is the Logarithm of the 
Area. 

kxatn. Let the longed Side AB=:i9,5^ and per* 
pendicular Height BE^6,o7 

Then to the Log. of the Side AB=:ii9. 5=1. 2960^5 
Add the Log. of the Perpend. BEii^6,o7=o. 783 1 89 



Log.ofthe Areaofthei ^,,h ^r:. ^ r^^^r^^A 
[lomboides } ==118,365=2.073224 



The 

Rhomboides 



6. To mafare j Tri a nom ABC, Fig. X Vl. 

The Sum of the Logarithm of the Bafe, and of 
half the perpendicular Height (or vice verfd) is the 
Logarithm of the Area. 

Exam. Let the tifeife ]ic==:65,25 5 and the per*- 
pendicular Height AG=2i,5 j then f AGssio./f. 
Therefore, 

To the Log. of theBafe . . . BC==:65,25=t.8i45?d 
Add theLog.of half thelPcrp. fAG=r: 10,75=1 .03 1 408 

The Log. of the Area . . . .=4=751,4375=5=2.845988 

7. To meafwri tf Cjrole ABCD. Fig. XVU. 

f D=:the Diameter. 
In order to this, let] Pacthe Periphery. 
C A:t3the Area. 

Then the Rule^ or Theorems for finding thofe ft- 
veral Parts are as follows ; 

E t Thco, 



2IO' MENstritATiOR of Svperfictes and 

Theo. I. 3.i4i6D=P. TJi. II.o.7854DD=A. 
TheoJII. o.3t8?P =3>. Th.IV.o. Q7957PP.= »A. 
Thco. Wi.2732A=D. Th.VI.i/i2.5664A=P. 

8. Therefore the Diameter AB being given, fup.- 
pofc =s 20,15, to find the Periphery P? per 
Thcor. I! ^ 

TheLog. of the Periphery P=:63,'303,6?r.=:i. 801426 

9. Given the Periphery of a Circle P=63,303, f^r. 
- to find the Diameter Tii per Thcor. IIL 

The Log. of the Dianxcter D or AB=;20. 1 5= i . 3042 63 

10. Having the Diameter given, fiappo{e=2di5, 
to find the iSirp, A of the Circle ? per Theor. 11. . 

The Log. of the Diameter . . AB=20. 15=1.304275 
Multiply by . 1.. ................... , _ % 

The Produd is the Log. of DD=2.6o855o 

II. 



Ch. XIV. SoM iy hoGAtiiTHUSr ait 

11. Or thus fuppofing the Periphery given,=63,303, 

£g*f. per Theor. IV. 

To twice the Log. of P=63.303={ [sVifze 

Add the Log. of the conftant j ^^ ^^^^^^^^ 
Number > 

TheSumisthcLog.ofj^_ 8 8 ^^_2 6oj 
the Area ^o ^ ^' 

12. Tomeafurethe Ssctor. of a Circle ACB, 

F/ff. XVIII. 

The Sum of the Logarithms of the Radius^ and 
f the^rfi& (or of the Arch and half the Radius) is 
the Logarithm of the Jrea. 

Examp. Suppofe the Radius AC=i2.36, and the 
Atch AB IP, 1 2 i then fAB=5,ii. 
Then to the Log. of Radius AC=i 2, 36=1 .092018; 
Add the Log. of fAB=5, 11 =0.7075 70 

^ of d2'sl'?! ."^I AC'^«3.^59«=r.7995»8 

i». To meafure the Sbpm?nt of a Circle, «8 
AFBG, Fig. XIX. 

The beft way is to find the Centre C •, as by this 

Theorem, ^/z. TB -FG=N. then ^ ^CG; 

whence C is given ; and finding the Are» of the 
whole Seftor ACBG, (as per laft Article,) and the 
Area of the Triangle ABC, (per Art..60 tf th^ awer 
Area be fubdofted irom the former, it wiUJeave 
the Area of the Segment required, AFBG, 

Ee.2 . . '. - H. 



^It MEMSUitATioK of Superficies and ' 

14. To meafure a Spherical Triangle, ABC. 

Ftg.XSi. 

From the Sum of the three Apgles A, B, C, take 
186 riegrees \ then from the Logarithm of the Re^ 
puindcrfubduft the Logarithm of the conftant Num- 
ber 720 ; to that Remainder add the Logarithm of 
the Superficies of the whole Sphere ; the Sum ihal| 
be the Logarithm of the Area required. 

' Angles be YdS || 

I 

Their Sum is . • . . 190^ 45' 

Subftraft ....... 180 00 Log. 

There remains .... 10^ 45^= io>75 == 1.031408 

}Jubdu<5l: the Logarithm of 720 = 2 J57332 

8.174076 

TheSumistheLog.ofthei -^^ «« /:^>, \ «>,«., ^ 
AreaoFtheTrSngle.. }4»^=^3.623:^i.2735i5 

15. To meafiire an Ellipsis, as ACBD, Fig.XKl. 

The Sum of the Lo^rithms of the Tranfvcrfe 
Diameter A B, the Conjugate Diameter CD, and the 
tonftant Number 9.7854 •, is the Logarithiu of the 
Area. " • * . r 

Exam. Let AB=6i,6, and CD=44,4. 

' • rAB=6i,6i=i.78958i 

Then add the Logarithms of < CD=r:4 4,4=1.647383 

' " ' * - tN.0.7«54=9.895O9i 



^h. XIV. Solids by Logarithms. 2 j| 



16. To mcafure a Parabola, as ACD, Fig. 22. 

From the Sum of the Logarithms of the Bafe AB^ 
the perpendicular Height CD, and the Number 2, 
fubftrad the Logarithm of 3 \ the Remainder is the 
Logarithm of the Area. 

Exam. Let AB3=6i,6, and 0)2=44,4 ; as in the 
ElUpfe. 

rAB=r6l,6=rl. 789581 
Then add thffXogjarithms of < €0=44,4=1.647383 

Land N.'>'.=o,30i030 

3737994 
Subftrad the Logarithm of • . 3=:o.477i2i 

The Log.of the Area requiredsssi 823,36=53.260873 

17. TomeafurcanyRBGu larPolygon,F/j'.XXIIL 

In order to this the following Table will be verf 
expedient. 

^Numbers. 



A Table for 
the more 
ready find- 
ing the A-^ 
rea of any 
Regular 
Polygon. 



Sides.' 


Names. 


's^ 


Pentagon. 


6 


Hexagon. 


7 


Heptagon. 


8 


'Odhigon. 


9 


Enneagon. 


10 


Decagon. 


II 


Endecagon. 


12 


Dodecagon. 



1.72048 
.2.59808 
3.63896 

4.82843 
6.I8I83 
7.69421 

8.51425 

9-33012 



Then the Sum of double the Logarithm of the 
Side of the Polygon, and of the Number in the 
Table proper to it, is thp Logarithm of the Area. 

Exam. Let the Pentagon ABCDE be propofed, 
and let its Side be ABsi4»6« 



JS14 Mensuration (f^Superficm and 

Then twice the Logarithm of AB= 14., 6 i ^-^^4353 

^'1 1- 1^4353 
AddtheL(^.ofthe/tfi«/tfriVi«»^.i.72048=::0.235649 

The Log, of the Area required =5:3^6, 74=2.564355 

In the fame manner you find the Area of any o^ 
thcr Polygon mentioned in the Table. 

18. Of the MjPNSURATION ^SOLIDS. 

To meafure ^ Cube ABCDPGE, Fig. XXIV. 

Three times the Logarithm of the Side is the Loga- 

, rithpfi. of the Solidity. 
Exam. Lf(t the Side AB ==: 31.57 = 1499^75 
Multiply by 3 3 

Jhe Log. of the Solidity . • . 31464,81=4.497825 
19. To meafure a Parallelopipedon AD, 

'" ' fig.'YiyiN. ' *" 

The Sum of the Logarithms of the ^readtb^ 
T>eph^ and Lengthy is the Logarithm of the So- 
lidity. 

Exam. Let th|B Width AB=2 1,5$ ; the Length 
AG=3r,57 J and Depth GFa^|p,o3, 

rAG=r3;i,57=i.499275 
ThenaddtheLogarithojsof<AB— 211,56=1.333649 

jLGF^ 9,03=0.955688 

The Log. of the Solidity J_>- r ' r 00 r 

, Inquired / }=6i46,2^23=:3.7886,2 

20. To meafure a ^rism ABCDEF, Fig. XXVI. 

Firft find"thc Jrea of the Bafe^ wHether a Triangle^ 
Square, &c. Tfreii the Sum of the Logarithms of 

the 



Ch. XIV. Solids by Logarithms. 215 

the feid Aredy and Length of the Prifmy is the Lo- 
garithm of its SifUdity. 

Exam. Suppofe a Prifm of a triangular JBafe^ as in 
theFigure, then let its Area be ABC=70i,4375i 
and its Length 80=70,15. 

Then add the Loga-J ABC=7oi,4g75;=;:2.84598J 
nthrpsof both . . . c BD= 70, 15 =1.846028 

■ '■ * 

The Log. of the Solidity zz:49205,845&?r.=4,692oi 6, 

^1. TomeafureapYUAMiD ABCDE. i^;f.XXVIL 

Firft find the Jrea of its Ba/e^ Whether triangular^ 
quadrjmgular^ &c. then the Sum of the Logarithnis 
of the Area of theBafe, and 7 of th^ perpendicular 
Height, is the Logarithm of the 5^//ii//y. ^ 

Exam: Let the Priim have a quadrangular Bafe 

ABCDas:j42»25 ; and the pe'rpendicukr Height 

GE=:i8oi thcniGE=5:6o. i 

Therfifotre add the Lo- c ABCI>=:34z,25=2.534^3 

. garithm^of. .... A ; TGE=6or=iw778i5i 

The Log. of the Solidity . . . : . =205^5=4.312494 
< • ^ .■• . ' 
' 22. To meafure a C y l i n- d e li "AGBDEF; 

%.XXVIIL 

r 

The Sum of the Logarithnis of the Area of its 
Ifefe, and of the Height^ is the Logarithm of the 

Exam. Let the Area of the Bafe be 380, i5=AGBH ; 
i and ;the Height BD=5o,05. , 

r . l:hen add the Loga-cAGBH=38o, 15=2.579955 
t rithpasiof ...... A BD;==5o,05==i. 699404 



Log, of the Solidity^ . • =i9o'26,5075=:4.279359 

22. 



ti6 MENStTR ATiON of Supetjichs and 

22. To mcafurc a Cone AEBFD, ^.XXDC, 

The Sum of the Logarithms of the Area of the 
^Bafe^ and -f of the perpendicular Height, is the Lo- 
garithm of the 5^//rf//y. 

Exam. Let the Area of the Bafe be 100.75= AEBF; 

and the perpendicular Height €0=19,95 ; and 

therefore ■yCD=6,65, 
Then add the Loga-c AEBf'=ioo,75=:2,oo3245 

rithms of • ...... 1 TCD=:6,65=:o,82282a 

The Sum is the Log. of the 1 ^^ . ^^ ■ ^ ^ -^ 
Solidity .;....:.. .-, . .{66^,9875=^.826067 

23. To Ineafurc t^ Frustum of a Pyramid; 
Pig. XXX. 

If it be a iquare> D:±=Side of the greater Bafe AB. 
Bafeas ABCDJ d ==Side of the leffcr Bafe ER 
then put-. . . C D-~dr=y and H rsthe H eight GO, 

Then we have this Theorem Dd-j-i^^ x H=: the 

Solidity. 

Exam. Suppofe D=50, d==2i j and H=i05,6; 
tjhen D— dr=;xt=:29. 

rD=5o. .=1.698970 
Then add the Logarithms of jd =21 , .=1.322219 

tH=io5,6=2;023664 

The Logarithm of • . . . DdH=i 10880=5.044853 

.III I " I I ■■ 

The Logarithm of ■5)«fH=296o3,2=4.47i337 

To which add DdH=i 10880 



The 3um is . . . Dd+i*JfxH=i46483,2*=: the SoK- 
dity required. 

24. 



Ch, XIV. Solids by LoGAkiTHMS. 217 

24. But if the Sa fes of thfe Frufttiin \it txiangu- 
lar% th<? Theorem is Dd4-5^;vxo,433H=the Soli- 
dity. Again, if the Bafe be any of the regular Po- 
lygons ; then put N = the Number^ in Table at 
Art. k 7 . proper to the Polygon, and the Theorem 
will be U(l+i;ff;tf x NH^the Solidity. 

25. To mcafure the Fruftum of a Right Cone. 

Vig. XXXI: 

Let Dcsthe Diameter of the great Bafc, AB, and 
a =the Diameter CD of the leffer Bafe ; I)— d=:v, 
and H=Bthe pterpendicular Height, as before ; then 

the Theorem for the Solidity will be 5it±^^ 

or thus ^Ppiwxb.7854H=Solidity. 

Eicam, Let AB=i6^ CD±=i2, and GOssp: then 
D— d==x=4; andTXX==5,3': alfo Ddcsiga; and 
fo Dd4"T^^=i97>3f. 

Then add theLoga-C Ddil-^^Jr=i97>3'=2-295i99 

^thnisof....;.] 07854=f 9.895091 

C H= 9=0.954242 

The Log. of the Solidity \ . =1394,8704=3.144532 

2.6. To meafure a Globe or Sphere ACBD» 
F;^. XXXII. 

The Sum of the triple Lojgarithin of the Diameter, 
and the Logarithm or the conftarit Number 0.5236 
is the Logarithm of the Solidity of. the Sphere. 

S£xam. Suppofe the Diamete^ of a Sphere AB=: 

50,37. : 

F f Then 



ei8 Mensuration oj Superficies and 

C AB=5o,37ss:i.702i7i 

Then add the Logarithms^ 1.702172 

7 I 1.702172 

i o,5236=.9.7i8999 

The Log. of the Solidity 1 ^-^ «, 
of theVcrc f }«669i3.8«4.8255i5 

27. To meafure the Superficies of a Sphere. 

The Sum of Logarithm of double the Diameter^ 
and the conftant Number ^•1416, is the Logarithm 
of the Superficies of the Sphere. 

Exam,. Let the Diameter AB=50.37. 

f AB=5O.37±s:i.702i72 
Then add thcLogarithms of < 1.702 1 yz 

t 3,14165=0.4971.51 

The Log. of the Superficies . • •• 7970,76*3.901495 

28. To meafure the Segment of a, Sphere, 
Fig. XXXIII. 

r D=The Axis or Diameter .of the Sphere CD, 
Let j C=Half the Diam. of the Segment's Bafe EB. 
C HsaThe Height pf the Segment ED. 

Then we have the two following Theprems for find- 
ing the Solidity. 

. C Theor. L 3CCH+H^ xa5236=:the Solidity. 
^''^' I Theor.n.3DH*— 2H^xo.5236=the.Solidity. 

29. To meafure a Spheroid ACBDA, 
Fig. XXXIV. 

The Sum of the double Logarithm of the Leffer^ 
the Logarithm of the. Greater Diajtnetcr, and the 

conftant 



Ch. XIV. Solids by Logarithms. 419 

ftmfiant tJumber 0.5236, is the Logarithm of the 
iblid Content of the Spheroid. 

Exam. Let the leffer Diameter CD=33 j and the 
greater Diameter AB=55, 

Then the double Log. of . , . Cn=33{ =J-5j85J4 

The Lo^ithm of ........ , AB=55=i.74a363 

The Logarithm of . . . . , ^ . . . 0,5236=9.719000. 

The Log, of the Solidity . • =31361,022=4.496391 

30. To mcafiire the Segmen.t of a Spheroid. 

As the Soh'dity of the Sphere AFBE is to the So- 
lidity of its Siegment AGK ; fo is the Solidity of 
the Spheroid ACBP^ to its lilic Segment AHI. 

31. Xo^ meafare a Parabolic Conoid ACBD^ 

'" Fig.XXXY. 

The Sunx of the double Logarithm of the Dia- 
pieter of the Bafe^ the lA)garithm of the Height^ and 
conjiant Number 0.3927 5 i^ the Logarithm, of the 
SoIidityoftheO^w^; ' 

'Exam. Le^ the Diameter of the Bafe AB;=r55 •, and 
its Height Cp=;::33. 

Then the double Log. of , . .AB^ 55 \ =1 ;7to363 

The' Logarithm of . • .CD=3=33=i. 51^8514 

The Logarithm of 0-3927=9. 594061^ 

The Logarithm of the Solidity 39201,4=4.593301, 
. Ff2 32. 



^20 Mensitration ofSuperficki atfS 



32. Tomcafurc the Frustum of a ParabolI^ 
Conoid, F/jf. XXXVL * 

r D=Duimeter of the greater Bafe AH. 
Tothisfend, !et)d=Diameter6ftheleflerBafeCD. * 

C H=the Height of the Fr#f;» FE. 
Then we have this ! 
Theorem . 



*| DD+ddxa3927H=th«Solidity, 



33. To m^fure a Parabolic Spindle, 
F/f.XXXVil. 

The Sum of the double Logarithm of its Thick- 
nefs, th^ Logarithm of its Length, and the Loga- 
rithm of the eonftatit Nu^aber 0.4185.8, ^ t^e Lo- 
garithm of the Solidity of the Spindle. 

Exam. Let the Diameter of its greateft Circle^ or 
Thicknefs ABs=s43,45 j ' and the Length CDs:; 
50,075. 

Then the double Log. of 'AB:^43,45{ =1:637990 

The Logaritlftn of CD=50,075=i. 699621 

The Logarithin of ...... . .0.41888=9.622090 

The Log, of the Solidity . . s=:3;^599,6=::4.5g76gi 

34. Tomeafureany oftheFivE Regular Bodies, 
/7^. XXXVIII. 

For this purpofe the following Table is neceflary. 



Names. 



•^ r 



Tetrahedron 
•< Oftahedron 
Hexahedron 
IcoAhedron 



Superficies. 
1.73205 



" Solidity. ^ 
0.1 1 785 



>< 3.46410 ^ 0.47140 > 



6.00000 
8.66025 



J. 00000 
2.18169 



Dodecahedron J ^20. 64573 ^7,66312 



Exi 



'am^ 



ph. XIV* Solids by Logarithms; ^zi^ 

Exam. SuppofetheSideAB=:i2, of the Icofahdron. 

^ ' F/;f. XXXVIII. 

The Sum of the Logarithm of the tahularNum^ 
^erj and double LogariAm pf the Side is the Loga- 
rithm of the Superficies'; and the Sum of the Loga- 
rithm of the tabular Number j and triple Logarithm 
pf the Side, is the Logarithm of the Solidity. 

']rhusthedouWeLo& 9^ • • • A?===^M ^J'So!?! 
The Log. of the tab. Number . . 8,66o25=5?o.93753c> 

The Log. o{tiieSuper^cies:sszi24y^o&S8:sp2-c^9S^92' 

Again the Logarithm pf . . . AB=7][2=i.p79i8; 

' , 3 

3-237543 
Add the Log. of the tab. Numb. 2.1816933=6.338793 

The Log. of the Solidity . .=553769,9458=3.576336 

3^. Thcfe Propofitions are fufEcrentfor meafuring 
any Superficies or Solid Body in common UCc 5 and 
the great Eafe and Concifehels of performing the 
£ime by Logarithms, is abundantly manifeft from 
thefe Exampks, 



^ 



CHAP. 



i22 Mensuration i^/zVi 

9QOGOQPOOOC3Q gffl OQOOC3QOOOOQQ 

G H A P. XV. 

^he Doctrine of Mensuration applkdu 

Gauging, Measuring Timber^, tf»^ 

Surveying; ^A^w^/Z'^ Practical Use 

ef the Plain Scale ^WSlidikg Rule, 

for thefe Purpofeiy is clear fy explain d. 

i» TN tfic preceeditig Chapter you have the prac^ 
X tical Method of abfolute or general Menfuration 
of the Content of Superficies and Solids^ laid down in 
divers Propofitions ; my Bufinefs is here to apply 
that general Doftrine to pai^ticular Ufes as thole of 
Gauging y Timber - Meafure^ aod Surveying. For tho* 
i there fhew'd how the Bimenfions of Bodies might 
be exprefs d in Ntmbers ; yet fince thefe Numbers 
reprefent divers kinds of ^antities^ as Inches, Feety 
Tardsy Poles, Chains^ &c. at pleafurc, fo they are 
equally fubfervient to thbfe fevoral Arts above-men- 
tionM, and muft be reduced tpthofc Denominations 
or Meafures \yhich axe peculiar to each. 1 ihal;i begick 
therefore witK 

at. Gauging. 

In this Art the Dimenfions are taken in Inches, 
and Decimal Parts thereof (or muft be reduced to 
fuch v) and by Inches is here to be underftood folid 
or cutic Inches ; and that as well in fuperficial as^ 
folid Meafure. For though it be improper, geome* 
trically fpeaking, to afcribe Thicknefs to a Superficies^ 
the Gaugers always confider them as one Inch deep^ 
and; accordingly compute the fuperficial Content in 
Gallons, 'or folid 'Meafure. 

3- 



fch.!5tV. to GAuGiNb. zzi 

3. As Inches are the lineal Dimet^ons cfCfugers^ 
fo Gallons 3TC the common Qiiantities o^ their filid 
Meafitrtj of liquid Subjiances efpecially ; and Bi^els 
for folid dry Meafurej 2s Mah, Cohij &c. Now 
the ft andard Gallons and £i^^7, as ufed in England^ 
are as fbllows : the Jle or Beer Gallon sstiSz folid 
Inches *9 the 9^ne Gallon sac 231 ; the Corn Gall^ 
1=5268,^8; and the Corn Bu/hel tst 2150,42 Cubic 
Inches. 

4. From whence it follows, that, fuppofing the 
Dimenfions of the Superficies in the foregoing Copter 
taken in Inches^ the feveral Aredts^ or fuperpcial 
Contents^ will by the Gauger be underftood to be ib 
ttimy folid or cubic Inches ; which therefore if he di- 
vide by 282, 231, 268,8, or 2150,42, the Quo- 
tients will be the feveral Contends in the refpedive 
Gallons or Sujbels. For injianccj in Fig. Xll. the 
Side of the Square AB=:3i,57 Inches, and the 
Area was therefore found (Art. 2.) to be 996^66 
folid Inches. 

r 282 )996,66(=3,53 Ale Gallons. 
Confe- \ 231 )996,66(=:4,3i Wine Gallons, 
ijuentlyi 268,8 )996,66(=3,707 Corn Gallons. 

(.2i56,42)996,66(=o,46 CornBuftiels. 

5. But fince Multipliers are the Reciprocals of Di^ 
^iforSj therefore Unity divided by thefe Divifors will 
produce fo many Multipliers. Thus, 

282 )i.ooooo(=so.oo3546 the Mult, for A. Gall. 

231 )i.ooooo(=o.oo4329 the Mult, for W. Gall. 
26^ )i.ooooo(ssxo.oo3722 the Mult, for C. Gall. 
2 1 50,42) I .ooooo(=a:o.ooo465 the Mult, for C. Buih. 

Confequently, if any Area be 

r 282 "J f o 003546I the Area f A. Gall. 

Divided) 231 /or mnlti-1 0.004329 I will be J W.GalU 

by . . j 268.8 fplied by | 0.003722 rexprefied 1 C» Gall. 

12150,42 J L000046S Jin .... LC Buih. 

6i 



^. > 



y>+ Mf MSURATION applied 

6, In order to work thefe Dimenfionis by the ir- 
iijicial Line of NumbetSy either on the ffain Scaler 
or Sliding-RMle\^ put B=Breadth, L=Lenjgth, D 
=Depth or Thicknefs, G=Standarci Gfallqn \ and 
AssArek, arid $=Solid Content in thofc dallons^ 

(^c. Then for Superfieies; ^ = A 5 and fo LB 

» AG ; therefore G : B i: t i A. That is, the Ra- 
tio^ qr Logarithm of the Ratio of th^, Standard Gat- 
Ion to the Bfiadtbj iS equal to that of the i?^i//i' of 
the Length to the jfr^/? in Gallons. Wherefpfe fup- 
^ofing 8=31,57, L=4i,5 (as in thitPdraileiogram 
r\g. XlII. Art. 3.; if you fet one Foot of the Coni- 
paiic^ in 282, and extend the other to 31.57,' thajt 
Extent wili reachi from the Length 41,5 to 4 t¥ 
Gallons, Ale Mcafure. Or, on the Sliding-Rule^ 
fet 282" on the Rule to 31,57 on the Slider, then 
againft 41,5 on the Rule is 4 -n? on the Slider, the 
jlrea in Ale Gallons^ as before j and thus the Area 
is found in Wine or Corn Gallons. 

7: Again, Since AD=Sx i, therefdre i : A::D:S. 
That is, the Logarithm of the Ratio of Unity i & 
the Area in Gallons A/ is e^iual to {hat 6f the Ratio 
of the Depth (or Height) D to the Solidity (or Ca- 
pacity) S, in Gallons of the fame kind. Where- 
fore if 3=21,56, L:^3i,57y 0-^:9,03, as inPa- 
rallelopipedon, Fig. XXV. then 0=282 :B=s2i|56 
:: L=3i,57 : A=2/<y the Area in Ale Gallbns ; cori- 
fequently i : A==2 t^ :: 0=9,03 : S=2i 1%, the fo- 
lid Content in Ale Gallons. Where ejttend the 
Compafles from Unity to the Area 2 tV; the fame 
Extent will reach from the Depth 9,03 to tht folid 
Content 21-1^5 Gallons. Or, on tht Jliding Rule, fet 
Unity I on the Rule to the Area 2 1% on the Slidef» 
then againft the Depth 9,03 on the Rule you have 
21 rs the folid Content in Ale Gallons on the Slider. 
Thus the Method of finding the Content or Capa^ 
city of fquare or i^dlilinear Arecfs at Bodits in Ale^ 

fVinei 



' t 



Ch. XV. \ ^ Gaugino; ^ 225 

fTine^ dr 'Corn Gallons is exceeding jflain ahd eaf;^ 
by the Single, Gunter and Sliding- Rule. 
' 8. In cafe oi circular Area% lince tfiey are all in 
the Ratio ocf ,the'Squafes of their Diameters v and 
fuppofiiig the EMameter' of ^Circle i Inch, the Area 
will JDeQ,7 8^ P^drnal Partsof a Cubic Inchi. tHore- 
fpre having the Diameter of any Circle given in 
Inches, if the Square thereof be multiplied by. 0.7854,' 
tlife Prodaft-wtlH)e the Area df that Cii^cle in Cubic 
Inches. ' Let Diithe Diainetert)f a Circle, 3=0.7854, 
GidStari^dard Gall, and A==Area"bf the Circle in 
fuchGaflons, as^efbrc. Then i : a :: IJP'! DDa=Area 
in Inches; therefore DDa=GA*5 and fo G : a :: DD 

: A = Area in Gallons. But a : G :: i : — s: the 

Square 'of the Diameter of that Circle, whofe Area 

iOi WhBttfore fince DD = J A, 

<hereforJifthcSqu?rero.7854) iSz.cooofzr 359^^5 

,oftheDiameterDDjo.^854) 23i.oodo(= 294,12 

be:dividedby2b=/o7854) 268.8ooo(= ^iM 

. * • .(,07854)2i50,4200(=2737.9:2 

or the feveral Diyifors 359^05 ; 294,1^, Cs?r. then 
thcfe Quotients will be ( the . Ajrea of the. Circle in 
JlC;^ fFtnCj &CC. GaKoncL 

9, Or tha?,, fi»|cc DD x ^ =5 A j therefcre if 

the Square of the Diameter DD 

U U'VaX'-^^^ )0 7854(0,002785 

be. multiplied I ^^i )o.7854(o,oo3389 
, by^= i2688 )o.7854{o.oo2922 . . 
... . C2 1 56,42)0.7854(0, 0P00036 "\ 
^jfeveral Produfts lyill be the Area in Ale, Wine, 
(^c. Gallons, aS' before, Suppofe the Diameter of 
^Circle D=;50 Inches, then DI>==2500 ; and put the 

donflant Divifo'^ j =dd= 359,05 •, 294,12, Csfr. 

G g Tlien 



j^2$i Mei^v iA^lov Jif>plied 

Then' fuicc ;xDD=:ddA, therefore d^tPDj: pr- 
: A= the Area in Gallons. Therefore with the Cxjm-* 
pafles fet oiv Foot in 359 1O5 extend the other to 
a^QO, the fame Extent will reach- from i to 6 i^z^ 
the'^Gallpns in the Area^of the propofcd Circle. 

10. Otherwife thus-, put ^ =^=o.o62}f85 j' 

0.0023^9, (^c. (fee Art, 9.) then bccaufe DDdd 
= A X K ; therefore- "we- have i : DV}:4idiA i or 
i:dd:i DD : A. If. then jfou fet i cm the Rule ta. 
0.00278 on the Slider^ you will feea^inft ^500 omt 
Rule 6-^ on thp Slider^ the Gallons of Abin.the 
Area of. that Circle, j; and thus you find the Gallons 
of Wine, Corn, &c. both by tht Sliding- Ruk and 
Thin Scale. - 

11. .Again, putting H= Height. of aCylinder^ 
and X) = the Diameter of its circular Bafe ; allGo S 
= the folid Content -or Cipacity in. Gallons 5 thcii^ 
dd' DD :: Ti : A ::) H : S. Now let D = 50 Inches^ 
gnd H=i5, the Area of fuch a Cylinder; will be 
thus found by the SHiing-Rule, Stet J595O5 en the 
Slider, to 2500 on the Rule ; then' againft 15 on 
the Slider, is 104 oh the R:ule';,atxd fo m^y Gal- 
lons of Ale would fuch an hollow Cylihdcr contain. ^ 

12. But* fincc^d-'r I :: DD: A,^ (Art. 9.) there- 
fore d : I :: D : /A ; and for the feme Reafbn d 
: /*B ::*D : V"^. Confequehtly , if a Jingle line of 
Numbers be made to. Aide by a double aney if; you fet 
d on tht Jingle one to i on the double oi^e^ then againft 
D on the former ^ you have A on the tatter, Alfo; 
if againft d on the Jingle Line^ you* fet Hon the 
double one ^ then againft D on Xht farmer," is S on 
the latter. By this means therefore/ having only 
the Diameter and Height of a Cylinder^ the Atea m 
the Safe, and folid Content of the faid'Oflittd^r/ h 
immediately known. The Reafon of thS' Method 

here. 



Ch.XV. ' ito Gauging; ^ ^^^^ 

' hett tfeEyifred^ appcats from wiat has been altetcfy 
iaift in the Chapter of /^ri^i?*/<y/ 0^rif//<ww. . ^ 
'^ V3-'^ow d = 

V 1^9>05= j8,95l^yyhicli Numbersf ^^ Gall, 
^$/ 294,12= 17,15 ( are called the] Wine Gall. 
•: 342^24=:i8> 5f G^^^ jP(?i«/ijCprnGaH, 
^1^55737^2=52,32 J for [^Com Bufli. 

And accordingjy on the finjgte Lirte of Numbers a- 
^ gainft thofe Gduge pMfts there is a fniaH Stroke (^}, 
1^ whicfr are itt the Capitate AG, WG, MB, to 
figmfy tlrey are Ae Omg& Pmnts for -^/^ Gstlkns-^ 
JVine Gallons^ and A£j// Bufhels. 

.14, Thefeforehi the^nemcncioti'd Cylinder, whofe 
Height H was cb 'i5 Inches-; and the Diameter ci 
i$.s Bafe D== 56^ in order to find the Area of the 
Bttiev and Capacity ^ the Cylinder in Gallons, by 
the Sliding- Rule ; fet i on the Slickr to the Gaugt 
t^ini for Ale 18.9^5 i)(i^t\^% Jingle Line^ then againft 
the JDiamiBtcr D = 50, on me fir^e Lme is 6 A on 
the double Une on the Slider, the Gallonl^ of Ale in 
the Area jof the Bafe. 

• Secondly, Set tYi^Gauge Pm4 1 8. gs to the Height 
15 ott the Slider, then againft the Diameter 50 oa 
the ^)gle Line is 104=: Son the double XiUxe Or 
Slider \ therefore the. Capacity of the Cylinder is 
104 (^llons, Me-NUafure^ as before. 

15. Or thus, with the jfe^& Line' and Compajfes 
only.* Set onh Foot m thz Gauge Point 18,95, and 
extend the other toiD =e 50, this Extent turn*d 
twice over from i will fell on 6 A 3= A = the Area 
of the Bafe in Gallons jo( Ahsi The folid Content, 
in this Cafe, will fall beyond the fingle lizie, or elie 
might be performed in the £im6 manner. 

16. With the double Lines of Numbers, on the 
Slidiiig 'Rule, thus. Set the Gauge^Point 18.95 on 

^the Rule to the Diameter 50 on the Slider, and mark 

the Nunibcr on the Slider againft i on the Rule •, 

G g 2 then 



«d«8 Memvuation i^lied ^ 

then agiimft that Number on the Rule you E^ve 
6 ^fs == A the Area of the Bafe on the Slider. Alfb 
mark the Number on the Slider agsii^ft the Height 
15 on the Rule, then again^ that Number on tl^ 
Rule is 104 on the Slider, the Capacity of the Cjr- 
lindcr as before ; and thus both the Area and Soli.- 
dity are found at once fttting |hc Rule. The fatnac 
is performed by the double Line and Compaflc4 
thus; .Set one Foot in the Qouge P^nt 18,95, ^^ 
extend, the other to the Diameter 50, the (ame Ex*- 
tent will reach at twice from i to 6TWthe Area^ 
and from the tjeigbt 15 to ;i 04 folid Capacity in 
Ale Gallons, . .. , j 

* j;. In this manner niijiy Gaugt-Peints be found 
for right-lined Areas: for kt a^yfuch Area givea 
in Cubic Incb^^ be called a y then (hy Art; 6.)\ i x^ 
^s GA, and fo G : 4 :: t : A ; or thus G : i :: ^ : A ; 
therefore •& :!:: s/mi i/^. 

Lv^2 150.42=46.36=3 LBuffi.- 

Call thefe Gauge Poinds d ; then d :jj i/T: • a = 
the Area in Gallons. Alfo i: v^H:: v'«:^S = 
the Solidity in Gallons. Whence you. may obferve; 
that rigbt'line'd Areasj as well as circular dnes^ may 
be found in Galions, by the^^/tf-and douhk Line of 
Numbers Aiding by each ^ other ; but not the Soli- 
dity of fuch Solids, there htrag ikrie^erms of the 
four variable, in the Analogy, for that, 

18. But notwithftanding this, there is a Method^ 
whereby the Solidity or Capacity of Solids or Vefleb 
may be^Aind /without knowing the Areas at all) 
by the Brendth^ Lengthy and Depth, only 9 which 
call B, L, D ; and S = Solidity, and G = the Gallon 
or Bufiiel, as before. Now fince BDL:=:GS, thttc- 

fore BD>£:gs = Gx^xS; andfoGx^:B^ 



Ch.XV* , to Gauqimc}.. ^z^ 

:: D : S. But bccaufo j;^;^ thjB RedpflDod of I^ 

therefore the Line of Numbers on which you feek 
Xi muft he inverted i and becaufe the Logarithm of 
G is added, the Number G muft always begin the 
invertffd Lincj or be placed equal to i on the dire£l 
Une. Thefe things premifed, 'tis plain that if B oti 
the Slider be fct to L on the inverted Line ordered 
as before, then againft D on the direff Une you 
hatre S dn the Slider. 

19. For Example ; Suppofc a Ciftern be 80 In? 
ches long, 50 broad, and 40 deep* Quaere the Con- 
tent in Bufiiels ? 

Hert G= ^150^2, ahd for thePurpofe of Malf^ 
Gauging tfiere is on (6mt Rules an inverted Line of 
Numbers fix'd on one Side the Slider ;. bjsginning at 
2150,42 as before iaid ; then on fuch a Rule fet 
50 ss B on the Slider to 80 = L on the inverted 
Liiie, and againft 40 = D on the direff Line on the 
Rule, you have 74=S=the Number of Bufhels the 
Ciftern will c(Hitain. ' And thus you might proceed 
for Gallons, had you inverted Lines beginning at 
282, 231, 268;8, 

20. In gauging Ca/ks^ the principal Conffdcratfon 
is the Curvature of the Staves ; as A B D. Fig; 
XXXIX- which G«^^r/ reduce to /wr D^^r^^j, or 
Varieties^'' vizi 

Variety I. Thofc {Caifcs whofe Staves are njoft 
eurved or tent^ are cbnfider'd as the middle Zone or 
B-ufium of a Spberoidy fuch as Fig. XXXI V. 

Variety II. If the Staves are not quke fo much 
crcbing or bent^ the Caik is fuppofed to be the mid^ 
die Zone^ .or Frufium of a Parabolic Sfindlej a$ 

i%.xxxyiL 

Vfiriety III, When the Staves of Caflcs are. but 
very little/ ^«ry^i, they are reputed to^bc^in the 
Form of the Fruftums of two equal Para folic Co-' 

noids^ 



kjcr MENSUltXtioN appUed ' ' ' 

waids^ joim^d together it the: wideft 6ale9. ^ jl^/A 
XXXVL 
V Variety IV. When tic Staves are ibait from the 
f f^;/^ to die Head, as the prickVi lings Afi, BD^ 
for very nearly do) ihtn 'tis plain Xuch a Cafk con- 
lifts of the Frtf/iums of two equal right Gones^JSst 
together at the greater Bafes. Fig. lOCXUL 

21. The Calks being reduced to thefc ibur Va- 
rieties, if you multiply the Di^erence between the 
Head and Bung Diameters 



for the < ii/y^wctyy 





and then add the Produft to the Head Diam^er^ 
then that Sum (hall be a vman Diameter^ or thajt c£ 
a Cylinder^ whofe Height -and Capacity is 6(|ual 
to that of the Cajk^ as near as poffible. 

az. For Example •, Let there be a CaOt ADEG^ 
whofe Bung-Diameter BF = 3 1 .5 Inches, and Head^ 
XHameter AGr=24,5i then their Difference is BF 
— AGp:;. 

r r .1 (rJI^'^Jll^yThe mean Diameter 
Confequently\7xo.6=28,7 / ^f .^e Cylinder 
24,5+ ]7xo.65=29>05r equal tpthic#. 
17x0.7 =29.4 3 ^ "^ 

Having found the Areas (by Art. 8,9, 10, i2^)>t* 
be 2,2385 ; 2,2941, ^c. Ale Gallons j andfuppo- 
fing the Length of th^ Qiik 42 IncheS, ssHL. 

Then the Contents f2,23«5x 42= 94.0^3! 
according to the 3 2,29^1x4^= 9^-35l Ale 
feveral Varieties 7 2^3 504x42= 98.7x1 Gallons, 
will be : ; . . ,.{2.4073x42=101.103 

And 



Ch.XV. /fl T;MBER,-^KJiwm«: 131 

And thefe .being all the princtpal Articles, w^heron 
the yfe of ififtrumental Logarithms^ or tk^ artificial 

Sne af JsTumbers is diredlr conoerned, I fliaU fay 
m^ on^hc Head of Qau^ingy \^^ ]pr^^ed ta 

thp nejtt Articje of .... 



r-.:) 



X^mier^iJ^^furc]. 



2 J. Ev^ry Piece of ^mber is a Solid, l^kcto fiunje 
one ijf other of fhofe, whofe aifoluti Mensuration 
was ftewn in the JaU Chapter, viz. The Frujium of 
zCdnt^'VtyUnder^ tilt Prujium qf a Jpyramid^ zPa^ 
rknelopipedon', Prf/mj 8ci;.[ , -.,"?..'. 
' !24i In .w||iatever Form the Piece h^pens to be* 
find the aintint ox Solidity Jn I^hes, ^ there taught^ s 
tK^hich'divide by 1728, (the /olid Inches in on^ Foci 
dp Tinjibcr) the Sluodent is the/olid C^tftent 14 Feet^ 
But ^H^t^ the Analogies for Operation, by Inftni- 
jTiients^^^ky be evident,^ I ftall m^ ufe of the fore- 
going Method in Gauging^ by putting B = the 
Brcadfit,' D=pepth, t.== Length, G=i2, and 

S =*:^qiidity, aUiiii \m\9k. Then" fmcc ^= S, 
therefore J5I>L=:;G^Si but J^ = thic LrngthvoiFcet^ 

^hich.lct be F, then BDJi=GGS = BDF ^ and 

confequently GG : BD :: F : S. 

25. No* if the Piece of Timber be in form 
of a fquare Prifm^ then t£e Bafe BD' is % buar& 
NMmber^ which dall gg\ whence CGzfigr.¥:S \ 
and thefcf<we.G:f ::v^'F:v^S. Wliercfore, bar- 
ing ^Jingle afiid double IJjh of Nmbers^ by the Sli^ 
ding'RuJe^ fct G=: 12 on the^ Jngle Line^ to th« 
Length in Feet F on the double one; then againft 
g on thp Jingle Line is S = Solidity on the double 
Line oh the Slider. Example, Suppofe a Piece of 
Timber, the Side oJF whofe fquar« Bafe is Fm^ 

inches. 



ft§2 MENStrRATTio^r bppHed ' ' 

Inches, the Length i8 F^±: F; Quere thi folidl 
Contents? • ^ ....:. r 

: Set the dbnftant Point G = 12 ob thc/nghZine to 
the Length m Feet F r= x8 ; thenagainft the Side of 
the given fquare Safe |^ = 15 Inches, on the former, 
is 28 = S = the Number of the .folid Feet in the 
Piece, on the latter. Or with the Compafles, ox- 
tend from. 12 to 15 on ihc^ugle Line^ the iame will 
reach from i% to 2i8 =:^.S in the dpubjc Line, dje 

.26. But If you have no//i^/^ tine^ proceed with 
the double Line on the Sliding-Ruk uimx Set the 
conftant Number GG= i^ on the Ruje.to, the giveri 
Squ^e ^==2 2 5 on the Slider ; then 'agaihft the LengA 
inFeet Fr=:i8, oji the Ruleis^ =$ = tHe folS 
Feet on |the Slider. ^Vy with the Cqmpades^. ex- 
tend ifoia i44r ^0*225, the fame Extent^will jcacii 
fronj' 1 8 to 2 8 the Tolid Feet, as before. ' Otherwiie 
by the Analogy G t ^ :: ifF : v' S ; thus, fet thexon-. 
ftant Point G= 12, on thq Rule to the.- given Sidp 
^=15, oh the Slider, arid ^ mark 'the Point on the 
Slider againft 15 cntheRutei bi-ing that Pbint t6 
12 on the Rule ; then jig^r^ft F:^ iS.onJthcRule^ 
id 28 tzS oh the Slidevy the 5ip//W//y as before. But 
much better with the Cqinjjsifles,. thus ; J^xtepd from 
12 to 15, that Extent tui^iiM fwiec ovtt will teach 
from 18 to 28 =S, as befotc. ' : - ' ' ' 
• 27. fif thie Keccll^e ife the Form of* a Parallelopu 
ped&Hy t^at' is, hath its Breadth and D^pth uneqoal^ 
then the common : way vis ;tp: atdd iht .Depth and 
Breadth tQ^tthtt\ and to t^ke half thatuSvnn fen: the 

Side o^ a mem Square; */». ^^^ tssg^ and then 
they imagine ^^ = BD, andfo mcafure ^ihe Plege as 
before. But this is a very ' erroneous way v^ and the 
more fO| as the Difference between the Breadikh and 
pepth is greater. For fince, (in order to reduce thi^ 
Piece to a 'fquare Prifm) BD = gg\ 'tis evident 



fch. XV. to Timb^r-Me/A^WJi^: >a J3 

|rte£s i/BD i that is, 'd^Anean.Prtfporthnul betwefen B 
:^d D, (for B: VBD:: •BD: t)) knAyi^^xk^fbilf 

^ ^."«" J as they ighorantly; fuppoftl''' .\ 

28. For Example ; Suppofc a piece ofbewn Tim^ 

#rr • iaform as aroreiaid, 'whofi Bicadthis^a^J In- 

^-^hcs=±: B, the^ Depiif or 'Tbickaefi-to Itches as !>« 

* imd Lcn^h 18 Fceir ifc T j whit is th« f(jlid Con- 

.j: .2.9. Find ft wf/?/r Prpfsr^ansi bet^Vceti; 8=^2,5 ajid 

^ b = 10, (as heretofore taught) it (hall i)e ^. =± 15 

'i==V5S*i' and thierefore 6**= 144 :^*'22|l: Fd:i|8 

: S ='2S(,*jas»befofe. ' But accordiij'g .to the/comrti6a 

Vhich^s greattfr'tKah- the true Area^of the^Bafe BD 
<^'2r5 h^i^^x;>6i^fquare Iftehs/^ AKb f* Gzr I2 
•' 6iiith«>fwg^ LOU of ^^Numbers t6 theJjBngtfc F set 8 
'^f*ect «fa ttte dtMbl^:lhen^^auiff^Ikiilfe)^==j6,25 
<rP^i?be^/irwi^, is .32! a? S oathe ktt'er. _But 32! 

^— 2S = 4! Feet more than is reaQj J!,n 'theTiece^' 
-• ^"'igV-^Becaufe' :B*I>'F te G^S ;'- therefore BD = 

v^ Si \tid fo.wcihacve Q\k y.RiipTS., U .then 

you have 2Ln Inverted Line of Numbers begmnifig 
^'fK)i»^G*fc=: I4I pHacfed oh onfc (ide^lfcerSRder^.^with a 
-'j^irrlfXi^e bn the^bthir ;^ then mAy the filid Content 
^^n'Feet tie fotitid tjy the®, D,' andi^i as taughr^in 

Art;!?; hereof,- thus ; Set B «afti22,5 on theSHdir, 
• 'to^FsiiiiS; oif *he in*uiited^IJms then againft D 
rtei xo^: 00 thfidireii'JJnt k iS^^n S, on tm SUder, 
. theLfpKd Feetfas before, ^ 

. . . SO, if ,th« Timber be In JPorm oJF the Frujium of 
. a Square Pyramid^ a^ Fiff . XXX. the Theorem for ^ 
» it^ Sfrfidity in inches is HD^ 4" iH^-x, as pir laft 
; Chfips^.A^rt, 23.; Thercfore^HDd-f iHx?c = GfS 

(where S== Soliidity in c»^zV7^tff/)J and fince i'iH==F 

te the Lepgth in Feet,' therefore FE)d + ifxx = 
' H h G 



vSdp^ a f insfi df DHmbetaf Incbid (quare bt fJ»e 
gretteft end, 9 Inches (quare a( th^ lefEst Enid» .aod 
ao Feet l<Mig, how many (olid Feet is f Here in Tiich 

•^Tftef. . • ■• ; 
I! IieteF=£2o» Dd=s2\5,9ndi)ffi^=f8-5ti^.ihak- 

/fore D«I*HxK=gtOi,3"> wherefort G*asM.4dtJM»o 

• v^yio^^i : 4},i=sS, the t^bmher o£ foUd F«ec ^tir- 
quired. Note, the common way, by (uppofiigidie 

: Sqflafe of ^ W m^-i** is very ^fej iiA in 

<. this .Indance would not give* the Solidity alibve 40jf 
! Feet, which is threcTeet lels than the Ttuth._ 

• "31. If thiyi!Stsot{lAtPruM»i^'P^'^^lifH^^^ 
- Vs is^ the Cale 6f moft Fi^S^f i5«eV^/«AEr^;^Uiin 

• j^ey «ay be: reduced to Fniftun^.of yj«arf. i'jrw- 
.. mds^ thus 4 4-et,A:ss tije Area qf ^ g^tqr'Bai^, 
■' ft = Afflka o f : the l afler. i TAen, t< AaP*. «ti^jt<5a 
■ s-z/i aAdVAxTasn<^i *l fo:fince D**4- a B)i 
' 4. dd':=iiuc,y thetefore A^'2 y/Aka"4iai=ia<*»'iriaro 

r|Saif£ ^ '4i4|S3ii "oonfeqwntly, G* : F :: 

':.|!A V v'^A^H^-: $ te'the Nimbcr- of y&ir*? ^/ 
';-ih.fucb'a Ti5w., ,1 .;, -,. ••-'■•; . . ..:!•/• 
;: v,yL. F<m: Example j Sqpppfe:* Piecft_pf /qiBJrtd 
.fBxs&es'.h&jit.Whes broad^^^d'ap dwipr^vtte 
uJargeft End j -aftd.; i cl bf«>a4 a*?^ 6 6e&^at theil#r 
;Endi thcLei^th 18 Feeli. Qj??^ f^<t w4^otent? 
:: :Bpr e:A== 3? x z Q ae ^40 » and » t=. ip xjisPi^ J 
t- and" v^ Ax'a,6aV c^8400ateVi9<^959 » . there&re 

■J A + f Arira4- a =55 298,653. ^TherfG''''^1^44 
•:F= 18:: 2:9S;6^3; 3 = 37,^3 the folid 'Fftt re-' 




pfad; 
the common or cu 



Ch. XV. to Tjmber-Mb A8iPfR«, 2 J| - 

33. If the Timber be in form of a Cyfliodef, F^^. 
XXVIII. tihcti ^tting a = 0:7854, and G =p la r 
Inches, we have (as per Art. 8.) thfe Analpgy j/ 
i: a ::DD: DDa=the Area of the Cylindw^sfia^-iri 
Inches (iuppofing Dss Diameter in Inched i) Ic^ . 
Hiss Height ia Inches-, then HPDa =t the. Inches 
fo'lid, whereof 1728 se G» «: one /i//<i i^<w^ ,. Con-' . 
feqirtntiy, if S?s;folid »£«»/«»;( lr» Feet, BTsHDaj^: 

Ci*Si and S =aF=-,tJie Length in^ Feet,; there. - 

foreFDDa=5G*S j and FDD = ^ &i«n<i Pitting 

^st dd « 1 83,34 i then FDD = ddS » wherefore 

(i^ :'Dt) : : F : S, or d i D :: /P = *^ ' . ^^^ 

.. «4. Or with the Length' in Inches, to find tftfc, 
Stjpty in ]Feet, thus \ fince HpDj = G'5,. there-' 
fore putting ~ «^<^. weiballhaveHbD==^Si 
' cftnfequently 5i: DD' ::H': S ^Solidity iii Feet j 
or </ • D ■• yT5 ; *^*S. , Wherefore, if there be a 
fe/ijV of Numbers on your SHdinglltjlei ^ fetthe 
c&j^ht 'Numbe? d = 46.9 o« t^e^fingk |-inc..to: 
^ « the Height in Inches.; on the douUe Lineoii 
the Slider; then againft D « thi^ Dmnetsrrof the. 
Cvlinder»8 Bafe'in Inches ont\ittJngliU»«, is S :^; 
the Number of folid Feet on the^Ud«r'afr requj^d, 
Aftd if 'the Length be given p Fjet j then the bpr 
lidity ia found, as above. Art 13. . 

-35. 'If you have ho fmSrle Line on yont Rufe, yoy 
rouf work with the Anabg4«MiRP:;F:;S i or 
dd iDDiiHi S. In the firft dd = 183,34, and dd 
,« 2200,154 i theiefbredsFb 13.54 = Diameter of 
i Circle, w&en the Area is i44:i «»»<! d =?= 4«.9 =* 
AeDiameteiSofaCitcle, whofeAreais 1728 ^fthq 
Inches in a folid Foot. Andltfctfc Numbcts, bemg; 
cSh'flint itt all Opeikion*, and the I^Icthod of opei 
nitinS Cl^elvfcfeA'pt Analogies ttcry way o« the f r 
■t-.-F'^ Hha 'y*""^ 



tijicial Linis ofLogaritbmi already, fiiffideotly exam* 
plifkd in ^ proceeding Aftides ; I (hall noffaerp^ 
again re|)eat it. 

3d. The cominon way icf meafiiring round Tim^ 
beTj is l)y girting * them about the Middle with a - 
Siring^ and taking lof the Girt for the Side, of z 
Squats equal to a vtean circular Area^ fach as would 
r^uce the propofed Piece tx> aXj^/f»&r^ But this 
isLalfo very mfeat^d-ungeometricaL' For; (in^e the 
Area of that Circle, whofe Circumfefen^e is" i, is 
o.a795^ ; and-thc- Area of that Square, whofe Side 
is ,25 (=1 of the faid Circumference) is 0.0625, 
and the Solidftics bemg Ml proportion to thefe Num- 
bers, W2. as 0,07958: 0,0625 ; that is, as 2 3 to 18;. 
'tis evident,, the Content by this falfe way is above 
4 lefs than what it really' is 5 which Err or ^ if it be 
not confiderable enough to be regarded and correfted, 
is great pity indeed. i 

; yf.. Therefore to-mea&re.a-Picca of rmnd tape-, 
ring Timber truly, it muft^be confiUered as the Fruftum ' 
of ji rigbi. Conei vyh ofe. folid 'C ontent in Inches is 
found by the Tlifiorem 0^4-1x^x0.785411^ as/<r • 
Art. 25. of the laft Chapter. Now putting 07854. 
*tt'a,.and G' = 171B, the cubiclncnes * in a folitl' 
fbct; the Theorem will become Di 4";}*"^ xaH 
=: G^S ; and ag ain (ince i\ H =5 F, the Length in 
Feet ; therefore Vd + 4^^^ x a F tsG^S i alfo put 
$^ sss^d; and then 'Dd-^^ixxxF =zdiiSy where- 

^re we have dd I'Ua^yxx :: F : S =?? th^ folid Con^.: 
tent in'Feet.' 

■ 38, For Example; Suppofe a Piece of. round. 
Timber be 36 Inches Diameter at one £od, and 9 
Inclies Diameter at thei other ; and 24 Feet long j 
cuasre the iSolidity in Feet ? .^ i , 

^Heit 0=^:36; i/x=9, D— .i/wy=:.2'7, Prf«. 
324, ^*?«=ki'45i-I^ + "^^«567» Wd? = 24. 



Ch.XV. . /^ Measuring ;Lanp* -^ 237. 

Therefore the Anelogjr is as dd=: 1 83^34 : 56;? :? P 
=?; 24 : S :;='74,'22 the foljd Feet reqi^ircd ; a3 mayj 
be wrought by the Lines of Numbers in luiy of tho; 
bcfore-mentiotfd ways. * • *.- ' ' z ' 

0/* Measuring Land; >• : 

39. What Iprineipally defign here, is to fhew^ 
bow the true Area or Contetit of a IMoT'of Lftnd is 
to be found by the arterial Line of Numbers^ in 
^cres and Decimal Parts. The Dimenfions 6f a Field- 
are commonlv taken in Rods or Poles i each con- 
taining 16 f Feet. Of thefe Poles, 40 in Length 
and 4 in Breadth nuke an Acre ; or an Acre is zsz 
i6q fquare Poks. Some (and indeed moft) ufe ^ 
Chain, called Gunter^s Chain^ in taking Dimenfions^ 
which conHfteth of 100 Links, and the whole in 
Length = 4 Poles or Rods. And fo 10 of thefe 
fquare Chains make an Acre. 

4Q. The Field being meafur'd with the Pole, if 
it be in Form of a Parallelogram^ put L = Length, 

and B z=: Breadth ; and then it will be — s= A = 

'- 160 

the Number of Acre$ ; therefore LB = 160 A 5 and 
fo we have 160 : L :i B : A = the Acres. For Ex- 
ample» fuppofe a Field be 35 Pole broad, and 185 
Pole in length 5 how many Acres doth it contain ? 
Set 160 on the Rule to 185 on the Slider, then a- 
gainft 35 on the Rule is 40! nearly, the Number of 
Acres required. 

41 . If the Field be in Form of a Triangle, and 
meafur'd with a Rod or Pole ; then, having plotted 
it and .meafur'd the Bafe and perpendicular Height, 
which call B and H ; then B x f H = 160 A j and fo 
160 : B :: IH : A :;= the* Acres as before. 

42. If the Field be in Form of a Trapezium^ as 
Bg. XL. then becaufe, by drawing the Diagonal 
AB, it is reduced to two Triangles ACB, ADB, 

and 



3538 • MENSCRATrON^^/Sflft/, &C. 

and ealfing the two Pcrpcndkrulars Cr and W,* H 
«&d ii and the commdn Bafe AB^JB^ we have 

^H-^»H+TxlB = i6pA. Therefoii 169. 

) IB :: H 4- ^ : A =? the Acres contained in the .Tra- 
pezium ACBD. 

43. If the Field be of a mult angular Farm^ it nHift* 
when piottedt htt rediKed tp feveral ^rianghs apd 
trapnia% VfA then meaAir\i, as ;)rr Art. 4^4^- 
In csife you take the Meafiires with a Qhaixxof 4 Rods 
ilr I to Links^ then^ the Anabgies for Op^^tioA 
will be the fame as above» only inflead of 160 you 
miift iufe . 10 \ thus IQ ! L :: B 3 A. Art. 40* and la 
::B :: fH 5 A. Art. 41, fcfr. All which is fo ^fy ^« 
to nerd no Example, nor any thiag more tQ b^ ^vA. 
COdcemit^ i^^ 




^ C £[ A Pw 



, ^G HAP. XVL. , , :- 



other refpcfls it is very little. Bati'ih feding^tie 
Places, Diftances, ^c.-o^th^ h^venly^BoSes^hs 
Calculation of JEclipfes, t^c. tljey are Ter^^'.neco^^ 

"f6r'lfiiidihg the f rcyoi-tibnal Pstt-ts/-as;^tin,^ in fomc 
Degree, appeaf i)y the 'following Eramples. - 

' a.,\ E]f|b}^j}te :i ; A^mk,: the meun Idftoifm^, ^f Athe 

-^aad-JLogarithm of hiiJ>i&ince fron>4h<hEarth~? 

I' Mean A- c 4* f )Eq!aa^.i° J3'49^a^ 
nomaliest4 8 J tions C i 3236 J^ar.C 4.99.5501 

DifFerences i=$d' 



Thci^ fbr :tkt:TrbportmalJfarts.o£.tht EquB^oaii fcy 

by thc-tiOgiftical Logarithms, ^ 

• '/giye the jQiffisrence-, i 13 =;i()930 

. wMt«iye4rthe^?/^0pmly ,., . ,9r4fir?J*fe. ^^8 

Anfwer, thie prop. Part ..... o 59 =1^7878 

This futyftfe^ed, (b^caufe the ^fqu^tion is- dc* 
^^creafing) from the Equation i^ 32' 49^ (anfwcring 

I tht 



\,tt4o • f35^ PftAtTifcAL Use ^JfiW 

the Anomaly 4* 7°) leaves the true Equation :S* 
J® 32' 50^ as^was require. 

-** 3. Then for the' proportional Part of the Lbga- 
rithm (ay 

One Degree, oi»^ . ; % . .-. . / ; J ; . ;^ 60' 00^= o 
givy t^^ Diffcrcnccx>f lx>gu-ithms . ^ io65?i53ia> 
what gives the Anomaly . . . *. ^ .. . 148' 1 4^= 948I 

Anfwer^ the Proportional Part, i * . . * . ^5=16258 

^VhichTubdtt^ froni theXc^rithm 4,95560^, ah- 
. iwering-to the Anoip. 4* 7^. itere.remain54,95B522, 
' the true 'Logarithfjf of. the ^un*s DiJianc,fttom the 
. Eafth^ as requii^fd. : 

^. fixkihple. IL, Sdppbfc ' the Moon's anhual Af' 
^g^^^»/ bj5 29^ 51' 37*, what is thelEquiattion of the 
jipogee inSi ^tcmtrhUy Qi}itv Oxh^ ." \. 

- ' Amiual t^^ I Etfus*- J 8*^'$3^ &^ 1 Ecc«i- 1 6^392 
. Argum;t 30 5 tteii (9' 07 i4 i ttiikyV6to4s 

v.-' j ( .> v -> ) ::i ii i G Gill r .*^ .: > ■! ? ' * 

^Differences . i=;.6o', 14 6 ^ , 65i 

T^ ' . ^ ;^ r. ■ ^ i / 'iiii .1^ ii "] . } ' ' 1.1* III I I 

- Tlifcn for the JEquatbn of the Apogee^ 6y 

lif pUfiJOegree, ot* .^ ♦. . , ^ . : . . ^o' oq1= ' 6 ' ' 
give the Difference . • . * . . 14' 6^^=6289 • 
V wlwtJgiyqi:! .ri.. * . * * ^:y^V''^^t^ ^'^j^is: .6^4;' '^ 

Anfwer, the Prop. Part • • * . I2' 8^ 6943 
*^T9 Which^dti" the Equation 8^ 53' ' 8^,' kgrfeeirfg to 

<r:: ^ ' -^^ II I ■ * ' . , - "li thtaft* Arg* 

<^tt^^ j^hE'^uii^is . * w ..Vg"^ 6s^ 16^ ths trm E- 
ijrisf27/rOTi of the Apogee as required.. 

5, Thc;i for- the true Eccentricity, fiy ,» 

*. 1 1 



Ch. XVI. Logistical Logarithms. ^41 

If one Degree, or . , 60' 00^= 9 

. give the Difference of Eccentricity . . 653 =7414 
what will the Part .,...•• 51' 37 = 654 

Anfwer, the Proportional Part 562 =8068 

To which add the Eccentricity . . • 60392 

The Sum is the true Eccentricity . . 60954, as re- 
quired. 

6. In like manner the Equation of the Moon*? 
Center, Latitude, Inclination of Limit, and Loga- 
rithmic Diftance from the Earth may be found. Alio 
the fame things are in like manner foimd for any 
of the Planets, of whigh there needs no more Ex- 
amples. 

7. Ejcample III. In an EclipTe of the Moon, ad- 
,mit her horary Motion be 30' 31'^, and the Sun'^ 

2^ 27^ ; then the hbrary Motion of the Moon froni 
the Sun \^ill be 28' 4!^ : and fuppofe the Moon hath 
pafs'd the Sun by the Diftance 1^ 19' 4'''=4744'^i 
What is the Time requifite for that Motion ? Say 
thus. 

As the horary Motion q( the Moon C • ni 

fromtheSun { *^ 4*3300 

is to one Hour ; 60 0= o 

So is the Diftance pafs'd above 56' c , g_. 
&\ viz. . 1 ^ ^^7' 



-*» 



To the Time above 2 Hours, viz. . . 49' 00= 877 

Therefore 2^ 49' were pafs*d fince the /r«^ P/>p^/w«, 
gr Moment of the Eclipfe. 

Note, becaufe in this Cafe the Motion of the 

Moon from the Sun, performed ip i Hour, is 28' 

j^^ ; therefore p,6^ 8'' will be pafled in 2 Hours : But 

^e prefent I^iftance of the Moon frpm the Sun is 

• \\ 79[ 



Z4:Z ^^ Practical Use oftbe 

79' 4^, which bccaufe it exceeds the Table of J>r 
giftical Logarithms^ therefore fubduft the Motion of 
two Hours, viz. 79' 4^ — 56' 8^=22' 56^ 5 and this 
Excefs. of Motion will give the Excefs in Time a- 
bove 2 Hours ; as in the Example. And thus you 
proceed when the Diftance of the Moon from the 
Sun exceeds one Degree, or the Table. 

8. Ejcample IV. Suppofc, in a Lunar Ecliffey 
xht Semidiameter of the Moon be 15' 15^, and the 
difference between the Mootfs Latitude^ and Sum of 
the Semidiameters of the Moon and the Earth*s 
Shadow be 9' 01^' ; Quaere the Digits eclipfed ? 

Note^ the Semidiameter of the Moon is always 
fuppofed to be 6 Digits^ or equal to 6 Degrees^ 
Therefore fay. 

As the Semidiameter of the Moon 15' 15^= 5949 

Is to jSx pigits (P 00' oo'^'zuioooq 

So is the faid Difference , . . • . 9' i^'= 8231 

To the Digits eiclips-d 3^ 32' s^"z::,\^^%^ 

9. Example V. In a Lunar Eclipfe^ let the Scru- 
ples of Incidence be 30/ 17^^, and the horary Motion 
of the Moon from the Siin be 28' 47^ j to find the 
Time of half Duration ^ 

Here becaufe 28' 47'': 60':: 30' 17^: a fourth 
Number greater than 60', and fo confequently be- 
yond the Extent of the X^ible; therefore (as in 
Art. 7.) from 30^ ij" fubduft the Motion for an 
Hour, viz. 28^47^, and to the Remainder i' 30^ 
|ind the Tiinc thus ; , 

As the Motion . , 28' 47^= 3190 

Is to an Hour ' ... 60' 00^'= o 

So the remaining Scruples i' 30^=: 160 21 

To the Time 3' 8^=1283^ 

^ ' *"' The 



Ifch. XVI. LoGISlriCAL L0GA«lTHAiS. 24J 

The Tij^ne ibught therefore i^ 3' 8^; the doubte 
bf which is 2^" 6' 16^, the Time of the whole Du- 
ration of the Eclipfe* 

10. Example VI. If the diurnal Motion of the 
JSun be 59' 8% 'what is the Motion for 7 Hours 

Say, as one Day ...... .24^ 00' ob^rr o 

Is to its Motion ........ 59 8^= 63 

So is the Time • . . 7** 15' 0^=5197 

To the Motion therein • . . i if $2^=t526o 

. II. Example VII. If the mean Motion of the 
Moon in one Hour he 32' 5&\ how far doth fh« 
move in 17'' 45'? 

Say, as one Hour . . i* 00' 00^= 13 802. 

Is to Its Motion .... 32' 56^=^:20392 ^ . 
So is * * . . 17'' 45' 00= 1309 l^d« 

\ 21701 

To- the Motion required .,9^ 44' 00^^= 7899 

Thus the Motion in 24 Hours, or one natural Day, 
will be found 13° 10' 35". 

12, Th^ Logiftical Logarithms may alfo be ufed 
with common Logarithms. Thus fuppofe you would 
find the Logarithmic; Sine of 18^16^47'', proceed 
thus > ' '* 

The DifFerences . . i'=6o" 383 

\ • i'i 2 ■ ' . Novf 



244 ^^^ Practical Use of the 

Now the Proportion is, 60' : 383 :: 47 : the Pro- 
pottional Part. But fincc the Logifiical Logarithms 
of 47^ and 60^ are reciprocally as thofe Numbers, 
and that of 60 is nothing ; therefore (the Differences 
of the common Logarithms being proportional to the 
Numbers 60^ and 47^ fay. 

As the Logiftical Logarithm of . . • .. 43^= 1447 
is to the Logiftical Logarithm of . . . 60 = o 
So is the common Logarithm of . . . 383 =2.5998 

To the common Logarithm of the 1 q « ^ r - » 
proportional Part I ^^5 =2.45d« 

Which add to the common} ^ 

Logarithm . ; } 9.496154 

The Sum is the Logarithm • . .9.496439 of 18^ 16' 
47'^ as required. And thusjproceed in any other Cafe 
of like nature. ' ^ 

13. The Logiftical Logarithms may be ufed like- 
wife with the Logarithmic Sines and TangentSj as in 
finding the Paralkxes of the Planets, &?r. Thus 
fuppofe the Horizontal Parallax of the Moon 55' 
1 2" i the /ingle of her Ori with the Horizon 22^4'; 
and her Longitude in her Orb from the Horizon 
81^ 27', to find her Parallax in Longitude. 

From the Logiftical Loga- 7 o - -/ ^^n_ .g. 
rithm of the.Hor. Paral. r^ ^^ ^^ - 3^^ 

rSiae of theV ^ ^ ^ ^\-^.q 

Subduft the) Angle . . . T^ "^ "" = ^'^^^^ 

Sum of the j Co-Sine ofi^, ^^ ^ ^,*,r»^ 

[Lo«gitvde..r' ^7 0=9^ 

viz. 18.7470 

ThercrcmainstT.cLogif.Lo-} o ,i ^n^ ^^^^ 
gar. of the Par. in Long. ..I ..? a y 

14. 



Ch. XVL Logistical Logarithms* 245 

14. Becaufe when the Angle of the Moon's Orb 
with the Horizon is greateft 67° 14' 20^', and (hein 
the Horizon, her greateft Parallax in Longitude 
will be then ; and will be found as in the laft Ex- 
ample, thus; 

■ I ' MM 

The Sine of . . 67° 14' 20"= 9.9648 

Radius 90° o o =10.0000 

^IfL^g'^f!"!^;^"'}""" 5«' 37"= .8o3« 

15. To find the Moon's Parallax in Latitude. 

Subduft the Co-Sine of the j 

Angle of her Orb with > 22^ 4' o^'zsp.pS^p 
. the Horizon 3 

there remains the Logiftical 1 ^^q ^,/ // ir 

Logarithm of .... . \ ^ 5^' 9"= 693 
which is the Parallax of Latitude fought. 

16. To find the greateft and leaft Parallaxes in 

Latitude. 

'ji^htSb'ssrr }9° 4.' 4o"=,.„37 

Logiftical Logarithm of thei o ^^i 21//- , -,,8- 
greateft Parallax V °° 3^ -^77^:, 

Agwn, 



246 fbe 1»RACTICAL UsfE, ^C 
Agwn, from the L. Lmt. of 1 / ^«//— a - 

Subftraft the Co-Sine of the i 
grcatcft Angle of Orb with > 67^ 14' 20^^=9. sSy 6 
the Horizon ^ ' 



Logiftical Logarithm of thee o .,/ j^// ^.^^ 
leqfi Parallax 1 ^ ^^ *^ ^^^^ 

17. By thefe Inftances the Reader will be ap- 
jpriz*d of the great Ufefalneffe and Expediency of 
Logiftical Logarithms in his Aftronomical Calcula" 
iions \ and as the common Logarithms are laid on a 
Rule^ fo likewife are thefe; and fuch a Sliding- 
Rule of Logiftical Logarithms may be very ufeful 
to thofe who defire to be more expeditious than exaS 
in their Calculations. 




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LOGARITHMOLOGT. 
PART IL 

CONTAINING 

A Threefold TABLE or CANON 

O F 

LOGARITHMS 



I. A Canon of LOGARITHMS for NxTvuAt 

N V M B E R s from i to loooo. 

II. A Canon of LOGARITHMS of SiMssand 
Tangents to every Degree and Minute of the ^adrtmt; 

III. A Canon or Table oi Logtjiicd LOGARITHMS. 

All which are digefted in a New and moji Compendious 
Form, and very corred. 



LONDON: 

Printed in the Year MDCCXXXIX. 



fa I 



A TABLE of LOGARITHMS of 
NATURAL NUMBERS from i to loooo, 



w 





I 

oooooo 

041393 
322219 
491362 
612784 

707570 
785330 
851258 
908485 
959041 


2 

301036 
079181 

342423 
505150 

623249 
716003 
792392 
857332 
913814 
963788 


i 


4 
602060 
146128 
380211 
53 '479 
643453 
73*394 
806180 
869232 
924279 
973128 
017033 
©56905 
093422 
127105 
158362 
187521 
214844 
240549 
2648 I 8 
2878O2 


o 

I 

2 

3 

4 

1 

7 
8 

_9 

lO 

II 

12 

'3 
'4 

\l 
«7 
«8 

12 

zo 

ZI 
22 
23 
24 

2? 
26 

«7 

28 

!2 

30 

3« 
32 

33 
H 

11 

37 
38 

39 


oooooo 
301030 

602060 
698970 

7781st 
845098 
903090 
954242 


477121 

M3943 
361728 

& 

724276 
799340 
863323 
919078 

968483 


oooooo 

041393 

0791 81 

"3943 
1461 z8 
176091 
204120 
230449 
255272 

!Z?754 
301030 
322219 

342427 
361728 
38021 1 
397940 
4«4973 
43 "364 

447»58 
462J98 

477I2' 
491362 

505^50 
518514 

$3 '479 
544068 
556302 
568202 
579784 
591065 


004321 

045323 
082785 
117271 
I492I9 
178977 

206826 

232996 
257679 

281033 
303196 

324282 
344392 

363612 

382017 
399674 

416640 

432969 
448706 
463893 

478566 

492761 
506505 
519828 
532754 
545307 
557507 
569374 
580925 

51; 21 77 


008600 
049218 
086360 
120574 
152288 
181844 
209515 
235528 
260071 
Z83S01 


012837 
053078 
089905 
123852 
155336 
1 8469 1 
212188 
238046 
262451 
285557 


30^351 
326336 

346353 
365488 
383815 
401400 
418301 

434569 
450449 

465383 
480007 

494155 
507856 
521138 
534026 
546543 
558709 
570543 
582063 
593286 


307496 
328380 
348305 

403120 
419956 
436163 
45 1786 
4^6868 
481443 

495544 
509*202 

522444 
535294 

547775 
559907 
S717-J9 

583 •99 
5943:^3 


309630 

330414 
350248 
369216 

387390 
404834 
421204 

43775' 
453318 
468347 
482874 
496930 

s 10545 

523746 

S36ss«i 
549003 
561101 
572872 

58433' 
595496 



A a 



♦4 Thi Logarithms rfNaU Numbers to 400. 



o 

1 
2 

J 

4 

I 

7 

8 
9 
10 
II 
12 
13 
«4 
«5 
16 

«7 
18 

12 

20 
tl 
Z2 
23 

^i 

26 

*7 
28 

!9 
30 
J« 
3* 
33 
34 
35 

h6 

37 
38 
39 



69^70! 
176091 

397940 
544068, 

740363 

,et»9i3 

875061 



9*94« 9. 934498 



97773^4 



021189 
060698 
096910 
130534 
161368 
190332 
217484 
243038 
267172 
290035 



31I7S4 

33243^ 
352182 
37«o68 
389166 
406540 
423246 
4393S3 
454845 
469822 



484300 
4983 n 
5U883 
5*5045 
537819 
550228 
562293 

574031 

585461 

1 596597 



778iS» 
2041 so 

4»4973 

748188 

8 « 9544 
880814 



982271 



025306 

064458 
100370 

'3^539 

*643S3 
193125 
220108 

2455^3 

269513 

292256 



313867 

334454 
354108 
372912 

390935 
408240 
424882 
440909 
456366 
47"9» 
485721 
499687 
513218 
526339 
j3907b 

55H50 
563481 
575188 
586587 
59769s 



J_l _!__ 



845098 ^5090 

230449 - 

431364 
56S202 
67^098 



755875 
826 




029384 
068186 

103834 
136721 

167317 
195900 

222716 

247973 
271842 
294466 



315970 
336460 
356026 
374748 
392697 

409933 
4265 1 1 
442480 
457882 
472756 



487138 
501059 

5H548 
527630 
540329 
552668 
56^666 
576341 
5877U 
598790 



•55272 
447158 

579784 
681241 

763428 
832509 
892095 

944485 
99 1 226 



033424 

071882 
107210 
139879 
170262 
198657 
225309 
250420 

296665 



318063 
338456 
357935 

394452 
41 1620 

428135 

444045 
459392 

474216 



488551 
502427 

515874 
528917 

541579 
553885 
565848 
577492 
588832 
599883 



954242 

278754 
462398 
591065 
770852 
690196 
838849 
897627 
949390 
995635 
037426 

075547 
11059a 
143015 
173186 
201397 
227887 
252853 
276462 
298853 



320146 

340444 
359835 
378398 
396199 
413300 
429752 
445604 
460898 
475671 



489958 
503791 
517196 
530200 
542825 

567026 : 
578639 
589950 
600973, 



Tie Logarithms of Natural Numbers t& 1400. 



90 
fi 

92 

n 

94 
95 
9^ 

97 

too 

lot 
102 
10^ 

104 
105 

107 
108 
109 

no 

HI 
llSk 

119 
114 

116 

"7 
118 
1x9 

120 
121 

\zz 
129 

124 

1*5 

126 

127 
|28 

129 

130 

131 

133 
134 

135 

13^ 
«57 
f3S 






I 


954»42 


472s 


9041 


9518 


96.378« 


4260 


84«J 


«9S0 


97.3121 


3590 


7724 


8183 


982271 


2723 


6772 


7219 


99.1226 


■6^9 


$«3^ 


6074 


oaoooo 


0434 


4321 


4751 


8600 


9026 


Ol.»«37 


$259 


703? 


7411 


02.1189 


1603 


5306 


S7I5 


93»4 


97R9 


03.3424 


3826 


7Ai6 


7825 


04.1393 


1787 


5323 


S7I4 


9218 


9606 


o<.3078 


34^3 


6905 


7286 


06.0698 


1075 


4458 


4832 


8iS$ 


8$57 


07.188* 


2250 


<U7 


5912 


91S1 


9543 


08.2785 


3144 


61615) 


6716 


990$ 


0158 


09.3422 


3772 


6910 


7257 


10.037 1 


0715 


3804 


4146 


7210 


7549 


11.0590 


0926 


3943 


4277 


7271 


7603 


I2 0S74 


0903 


3852 


4I7» 


7105 


74»9 


X3.0334 


0655 


3539 


3858 


6721 


7037 


9879 


0194 


14.3015 


3327 



2 

5206 

^995 
4.731 
9416 
4051 
8^37 
3«7^ 
7666 
2111 
6512 

086S 
■5180 

94s J 
3680 

7868 
9016 
6124 

0195 

4127 
8223 

2l8l 

6105 

9993 
3846 
7666 
1452 
sao6 
892S 
2617 
6276 
9904 

3503 
7071 
061 1 

4192 
7(S04 
io'>9 
4487 
7888 
1262 

4611 
7934 
1231 
4504 
7752 
0977 
4x77 
7554 
050^ 
3<^39 



5^88 
0471 

$202 

9882 
4$I2 
9093 
1616 
8113 
2553 

1301 

5609 

987^ 
4100 

8184 
2428 

<^S33 
0600 

4628 

Uzo 

2575 

6495 
0380 

4230 

8046 
r829 

9298 
*985 
6640 

0266 
3861 

7426 
0963 
4471 
7951 
1403 
48^8 
8227 

il?8 

4944 
8265 
1560 
4830 
8075 
1298 
4496 
767c 
0821 

395 » 



_4 

6168 

0946 

5672 

»347 

497^ 

9548 

4077 

8559 

2995 

73«£ 

1734 

503S 

0300 
4520 
8700 
2841 
6942 

lOOl 

5029 

9017 

2969 

6885 

0766 

4613 
8426 
2206 

5953 

95^8 

3352 
7004 

o526 
42.9 
7781 
1315 
4820 
8297 

»747 
5169 

«565 

1934 

S178 

859^ 
t888 

515^ 

8399 
1619 

4814 

7987 
1136 
4263 



__5 

6649 
I421 
6142 

0812 
5432 
000 3 
45^7 
900 s 

345^ 

7823 

116 

6^66 

0724 

4940 

91 16 

3251 

73')0 

140S 

54iO 
94H 



3362 

7275 
1152 

499^ 
88u5 
2582 
6326 
0038 
37'8 
7368 

0987 
4575 
8136 
1667 

5109 



2090 
5510 

8903 

t270 

5610 
8926 
2216 

5481 
8722 
1939 
5133 
8503 
1450 

4574 



7118 
1895 

6611 

!27^ 

589« 
0457 
4?'?7 
9450 

3«77 
8i59 
259^ 
<5894 
1147 
53^0 
9552 
l66^ 

7757 
1812 
5830 
9811 



7^07 
2369 
7080 

1740 
63S0 
09)2 
54'-^^ 
9895 
4317 
8^95 
3029 
7321 
1570 
>779 
9947 
40/5 
8164 
2216 
5229 
0107 



3755 
76^4 
1538 
5378 

9185 

2958 
6699 
0407 
4085 

ZZil 
1347 
4934 
8490 
2018 
1^518 
8990 
2434 
585 J 
9241 
2605 

5943 
9Z5<5 
2543 
5806 

9045 
2260 

545 J 
8618 
1763 
4885 



J^\.^ 



4148 

8053 
1924 

57^0 

95^3 

3333 

7071 
077^ 
4451 

8094 
1707 

5191 
8845 

2^70 
66 

9335 
2777 
^191 
9578 
2940 
6176 
9586 
2871 
5131 
9357 
2580 
5768 

5934 
2076 
5196 



8086 

2843 
7548 
22<y 

6808 
1365 
5875 
0339 
4757 
9130 



3460 

7748 
1993 

6197 
0351 

448<^ 

8571 
£619 
66^0 

0602 

4540 
8442 

4309 
6142 

994* 
3709 
7443 
1145 
4816 
8457 
2067 

5<547 
9198 
2721 
6215 
9681 
3119 

<553i 
9916 

3275 
6608 
9915 
5.198 
6456 
9^90 
2900 
6o%6 
9249 
2384 
5507 



85M 

8or6 
2fttftfj 
72^6 

18x9 

07S3 
$19^ 

95 £5 
3891 
8174 
i4i5 
661^ 

0775 
4896 

8978 
30&1 
7028 
0998 

493* 
8830 
26^4 
6524 
C320 
4083 
7814 
i5»4 
5181 

8819 
2426 
6004 
9552 
3071 
5562 
0016 
34^2 
6870 

0253 
3609 

6940 
0245 

3525 
6781 

GO 12 
3219 
6403 
9564 
2702 
5818 





lie Logarithms of Natural Numbers to 1900. 


*7 
9 


N».' 





I 
6438 


2 

6748 


3 

7058 


4 
7J«7 


5 1 

7576 


6 

7985 


7 
8^94 


8 


140 


14.^128 


8603 ^99^ 


141 


9219 


9527 


983 s 


01 42 


0449 


0756 


1063 


1370 


1676 


i98£ 


143 


i^.i288 


2594 


«900 


3205 


3 J 10 


3815 


♦ "9 


4424 


47^8 


5032 


143 


OS* 


5640 


<943 


<S24<S 


^549 


6852 


7154 


7457 


7759 


8061 


144 


8362 


S664 


896 5 


926^ 


9567 


9%8 


0168 


0468 


0769 


^068 


'4") 


1^.1368 


1667 


19*7 


2266 


2564 


2^53 


3.6. 


3460 


5757 


4055 


146, 


4353 


4550 


4947 


524* 


554« 


5838 


61 34 


6430 


6714 


7022 


»47 


7J17 


7613 


7908 


8203 


8497 


8792 


9c85 


9380 


9674 


99« 


148 


X 7.026 1 


^555 


0848 


1141 


»434 


172^ 


2019 


2311 


2603 


2«95 


149 

15© 


ix%6 


347« 
5381 


37<9 
6670 


4060 
6959 


4351 
7248 


4641 
7536 


4931 
7825 


52X& 

8113 


55U 
8401 


$802 


6091 


86^9 


«5« 


8977 


9264 


9552 


9839 


0125 


0413 


C699 


0985 


1272 


»55« 


152 


18.1844 


2129 


2415 


2700 


2985 


}270 


3554 


3839 


4123 


4407 


»53 


4^91 


4975 


i^l^ 


5542 


5825 


6io8 


6391 


6674 


^956 


7239 


»U 


7521 


7803 


8084 


%166 


8«f47 


8928 


9209 


9490 


9771 


0051 


J5$ 


19.0331 


0611 


0892 


1171 


145 1 


1730 


2010 


2289 


25<2»7 


2846 


»5tf 


3125 


3403 


3681 


3959 


4*37 


4514 


479a 


5069 


5346 


5525 


>57 


5909 


5176 


6452 


6729 


7005 


7281 


755<5 


7832 


8107 


8382 


X58 


8657 


893* 


9206 


9481 


9755 


0029 


0303 


0577 


08 $0 


IJ24 


IJ9 

i6o 


ao.1397 


1670 
4391 


1943 
46^2 


2216 
4933 


2488 
5204 


275 J 
5475 


3033 
5745 


3305 

6016 


3577 
62B6 


384^ 


4110 


655^ 


idi 


6%p6 


7095 


7365 


7634 


7903 


8172 


8441 


8710 


8978 


9247 


i6» 


9$»5 


9783 


0051 


0318 


0586 


0853 


1120 


1388 


16S4 


1921 


163 


2I.ZI88 


2454 


4720 2986 


3252 


35'8 


3783 


4048 


4314 


4579 


1^4 


'4844 


5109 


5373 5'i38 


590a 


6166 


<S42o 


^594 


6957 


7221 


i<55 


7484 


7747 


8010 8273 


8535 


879? 


9060 


9322 


9584 


984^ 


I6tf 


22.0108 


0370 


0631 


0892 


1153 


'4'4 


1675 


J936 


2lp6 


2456 


i«7 


2716 


2976 


323<5 


3494 


3755 


4015 


4274 


4533 


4792 


50$i 


U8 


$309 


5568 


5826 


6084 


6342 


6600 


6858 


7II5 


7372 


7630 


169 


7881 


8144 


8400 


8557 


8913 


9170 


9426 


9681 


9938 


0193 


170 


2J.0449 


07Q4 


0960 


1215 


1470 


1724 


1979 


2233 


248 a 


2742 


171 


899(5 


3250 


3504 


3757 


4011 


4264 


4517 


4770 


5023 


5276 


17a 


15528 


5781 


6033 


6285 


5537 


678? 


7^41 


7292 


7544 


7795 


t73 


8045 


8297 


8548 


8799 


9049 


9299 


9559 


9799 


0050 


0300 


174 


24-0949 


0799 


1048 


"97 


1546 


»795 


2044 


2295 


2541 


2790 


»75 


J038 


3286 


3534 


3782 


4030 


4*77 


4524 


4772 


5019 


^266 


I7tf 


5513 


5759 


6006 


6252 


<5499 


<745 


6991 


7235 


7482 


7728 


"Z 


7973 


8219 


8464 


8709 


8954 


9198 


9443 


9687 


9932 


0176 


178 


25.0420 


0664 


0908 


"J* 


»395 


KS38 


1881 


2125 


2367 


2610 


179 


2853 


309tf 


3338 


3580 


J822 


4064 


4}0tf 


4548 


4790 


5031 


180 


5272 


55 '4 


V'^'' 


599<J 


62zi 


<J477 


<57i8 


6958 


7198 


7439 


181 


7679 


7918 


8158 


^398 


8637 


8877 


9116 


9355 


9594 


9833 


181 


itf.oo7i 


0310 


0548 


0787 


1025 


12^3 


1501 


1738 


197^ 


2214 


183 


M5I 


2688 


2592 


316a 


li99 


3<J3<5 


3873 


410P 


43^5 


4582 


184 


4818 


5054 


5290 


5525 


S7<5i 


59»« 


6tzi- 


6467 


^702 


(5937 


185 


7172 


7406 


7641 


7875 


8110 


nu 


8578 


8812 


904^ 


9279 


186 


9^n 


974<5 


99?o 


0213, 


044<5 


0679 


091a 


XI44 


1377 


1609 


187 


17. 184a 


2074 


2306 


2538 


2770 


3001 


3233 


34^4 


096 


3927 


188 


4158 


438914620 


485c 


1081 


53" 


5542 


5772 


6002 


6232 


189' ^4^2 ^6691 1^921 ' 71 SI ' 7?8o 


-2^ 


7«?,8 


^0^7 


8296 


8525 




i^^riir^^# 



\^ 


The Logarithm of Natural Numbers to 3400. "] 


/No. 





I 


2 


3 


4 


-1^ 


6 


7 


-.8 1 9 1 


290 


46.2398 


2548 


2697 


2847 


2997 


3146 


3296 


3445 


3594 


3744 


291 


3893 


4042 


4191 


4340 


4489 


4639 


4787 


4956 


5o»5 


5234 


29a 


5383 


5532 


5680 


5829 


5977 


6126 


6274. 


6^23 


6571 


6719 


293 


6868 


7016 


7164 


7312 


7460 


7608 


7756 


7904 


8052 


8200 


294 


8347 


8495 


8643 


8790 


8938 


9085 


9233 


9831 


9527 


9675 


295 


9822 


9969 


0116 


0263 


0410 


0557 


0704 


08s I 


0998 


1145 


296 


47.1292 


1438 


1585 


1732 


1878 


2025 


2171 


2317 


2464 


2610 


297 


2756 


2903 


3049 


3<95 


334> 


3487 


3633 


3779 


3925 


4070 


298 


4216 


4362 


4508 


4653 


4799 


4944 


J090 


iV'J' 


5381 


5526 


299 


5671 


5816 


5962 


6^07 


62^2 


6397 


6j42 


6687 


6832 


6976 


300 


7121 


7266 


^74<" 


7555 


7700 


7844 


7989 


8133 


8278 


8422 


301 


8566 


87 «i. 


8855 


8999 


9«43 


9287 


943' 


9575 


97 '9 


9863 


302 


4^.0007 


0151 


0294 


0438 


0582 


0725 


0869 


1012 


1156 


1299 


303 


H43 


1586 


1729 


1872 


2016 


2.59 


2302 


2445 


2588 


2731 


304 


2874 


3016 


3159 


3302 


3445 


3587 


3730 


3872 


4015 


4'57 


30J 


4300 


4442 


4584 


4727 


4868 


5011 


>'53 


$295 


5437 


5W9 


306 


5721 


5863 


6005 


6147 


6289 


6430 


6572 


6714 


6855 


6997 


307 


7138 


7280 


7421 


7563 


7704 


7845 


7986 


8127 


8269 


8410 


308 


8551 


8692 


8833 


8973 


9114 


9fi5 


9396 


9537 


9677 


9818 . 


309 


9958 


0099 


0239 


0380 


0520 


0661 


0801 


09ii 


108 1 


1222 


310 


49.1362 


1502 


1642 


1782 


1922 


2062 


2201 


2341 


2481 


2621 


3«« 


2760 


29<5o 


3040 


3'79 


3319 


3458 


3597 


3737 


3876, 


4CI5 


312 


4»55 


4294 


4433 


4S7* 


4711 


4^:50 


4989 


5128 


5267 


5406 


3'3 


5544 


5^83 


5822 


5960 


6099 


6237 


6376 


6514 


6653 


6791 


314. 


6930 


7068 


7206 


7344 


7482 


7621 


7759 


7897 


8035 


8173 


3«| 


8311 


8448 


85^6 


8724 


8862 


8999 


«"37 


9275 


94' 2 


9550 


3^6 


9687 


9824 


9962 


0099 


0236 


0374 


0511 


0648 


0785 


0922 


i^7 


50.1059 


1 196 


1333 


1470 


1607 


'744 


i88o 


2017 


2154 


2290 


318 


2427 


2564 


2700 


2837 


2973 


3109 


3246 


3382 


3518 


3654 


322 


379" 


3927 


4063 


4' 99 


4335 


4471 


4607 


4743 


4878 


5014 . 


320 


, S^S^ 


5286 


5421 


JSS7 


5692 


5828 


5963 


6099 


6234 


6376 


321 


6505 


6640 


6775 


6911 


7046 


7181 


7316 


745' 


7586 


7721 . 


322 


7856 


7991 


8125 


§260 


839s 


8S3C 


8664 


8799 


8933 


9068 • 


323 


9202 


9337 


947" 


9606 


9740 


9874 


0008 


0143 


«277 


041 1 


324 


51.0545 


0679 


0813 


0947 


1081 


1215 


IJ48 


1482 


1616 


175© 


32s 


1883 


2017 


2150 


2284 


2^17 


2SS' 


2684 


2818 


2951 


3084 


326 


3218 


335" 


3484 


3617 


375° 


3883 


4016 


4149 


4282 


44' 5. 


3*7 


• '4548 


4680 


4813 


4946 


5079 


52W 


5343 


5476 
6800 


C609 


5741 


328 


5874 


6006 


6139 


6271 


6403 


6535 


6668 


6932 


7064 


329 
330 


7196 


7328 
8645 


7460 
8777 


7592 
8909 


77*4 
9040 


7855 
9171 


7987 
9303 


9434 


8251 
9565 


828. • 
9697 


8514 


331 


9828 


9959 


0090 


0221 


0352 


0483 


0614 


0745 


0876 


1007 


332 


52.1138 


1269 


1400 


'530 


1661 


1792 


1922 


2053 


2i!>3 


2314 ■ 


333 


2444 


2575 


2705 


2835 


2966 


3096 


3226 


3356 


3486 


$616 


334 


3746 


3876 


4006 


4136 


4266 


4396 


452-6 


4656 


478^ 


4>i5 


33S 


5045 


5'74 


5304 


5434 


5563 


5692 


5822 


595' 


608 1 


6210 


336 


6339 


6468 


6598 


6727 


6856 


6985 


7114 


7243- 


73,2 


7501 


*^l 


7630 


7759 


7888 


8016 


8145 


8274 


8402 


8,- J I 


8660 


87s 


338 


8917 


9045 


9*74 


9302 


9434 


95S9 


9687 


9815 


9943 


00 ; 2 


^ii. 


53.0200 


0328 


04.56 


0584 o7t2lo84o'o96.8Uo95l 1223] 


liSJ 



340 

34> 
34a 

?43 

344 

346 
3.47 
348 
349 
350 
3$i 
352 
353 
354 

356 

357 
358 

359 

360 
36. 

362 
363 
364 
365 
366 

,3^7 

368 

369 

370 
371 

372 
•373 
'374 
'37S 

376 

377 
378 

37? 
380 

381 

382 

383 
384 

385 
386 
387 
388 

389 



fhe Logarithms of Natural Numbers to 2900. 



,..-r, , 1607 1734 1862 

2754 2882 3009 ^136 

4026 4f53 4280 4407 

5294 5421 5547 5674 

6558 6685 681 I 6937 



7819 7945 
9076 9202 
54.0329 0455 



1579 
2825 



4068 
5307 

6543 

7775 
9003 

J5.0228 

1450 

2668 

3883 
5094 



6302 

7507 
8709 

^9907 
50,iroi 
2293 

3481 
4666 

5848 
7026 



1704 
2950 



4192 

U31 
6666 
7898 
9126 
0351 
1572 



8202 8319 ^436 8554 
9374 949 « 9608 9725 
57.0543 0660 0776 0893 
1709 1825 1942 2058 
2872 2988 3104 3220 
4<53i 4147 4263 4378 

i\ i^^l 54^9 5534 

^34< 64.56 6572 6687 

74^92 7607 7721 7836 

8039 8754 8868 8983 

9784 9898 0012 0126 

58.0925 i©39 1153 1267 

2063 2177 2291 2404 

.3199 3312 3425 3539 

433» 4444 4557 4670 
5461 7573 5686 5799 
0587 6700 6812 6925 



^n 



8071 



9327 9452 9578 
0580 ' " 
1829 

3074 



4316 

5554 
6789 
8021 
9249 
0473 
1694 



2790 1 291 

4004 4r26 

52M 5336 
6423 6544 

7627 7748 

882S 8948 

0026 0146 

1221 1340 

2412 2531 

3600 3718 

4784 4903 

5966 6084 

7144 7262 



8197 



1990 
3263 

4534 
5800 
7063 
8322 



21 17 2245 



Jill 



0705 



1953 
3*99 
4440 



083010955 
207812203 
3323; 3447 
,,^- 4564:4688 
5678 I5802 15925 
6913 703617159 
8144 8266:8389 
9371 9494 1 96 I 6 
'^'■'^'" 0717 '0840 



5927 6053 

V^2 73*5 
8448 8574 

9703 9829 
1080 
2327 
357« 



0595 
1816 

3033 
4247 

5457 



6664 
7868 
9068 
0265 

H59 

2650 

3837 
5021 
6202 
7379 



88^! 7823 7925] 8047 8160 8272 



9279 9391 



8944 9055 91& , ,^ ^^,. 
9950 00 61 0173 0284 059 6 0507 0619 



1938 

3»54 



4368 4489 



5578 



6785 
7988 
9188 
0385 
1578 
2768 

3955 
5139 
6320 

7497 



8671 
9842 

lOlO 

2174 
3336 

4494 
5650 
6802 
7951 

9997 



0240 
1381 
2518 
3652 

4783 
5912 

7C'37 



2060 
3276 



5699 

6905 
8108 
9308 
0504 
1697 
2887 
4074 

5257 
6437 

7614 



8778 

9959 
126 

229,1 
3452 
4610 

5765 
,6917 
8066 
9212 



0355 



3518 
4787 



4812 
6049 
7282 
8512 

9739 
0962 
2181 

3397 
4610 
5820 

7026 
8228 
9428 
o6i4 
1817 
3006 
4192 
5375 
6555 
7732 

8905 
0076 

^243 
2407 

3568 



5880 
7052 
8181 
9326 



6024 6137 
7 "49 7262 



2372 

3645 
4914 

6179 

744« 
8699 



8 



9954 <^079 



1205 
2452 
3696 

4936 
6172 

7405 
8635 
9861 
1084 
2303 

35^9 
4731 

5940 

7146 
8348 
^9h8 

0743 
1936 
3125 

43", 

6673 
7849 



9023 
0193 

1359 
2523 

^. , 3684 
4726* 4841 



5996 

8295 
9441 



-_ 0469 0583 

1494 1608 1722 
263! -^ 

aIJ^ •'~'^' •'^^'' '**''^ 

4890 5009 $122 5235 



838^ 
9503 



8496 

9614 
0730 



2500 

3772 
5041 
6306 
7567 
8825 



1330 
2576 
3820 

5060 
6296 

7529 
8758 

9984 
1206 

2425 
3640 
4852 
6061 



7266 
.8469 
9667 
0863 
2055 

3244 
4429 
5612 
6791 
7967 



9140 
0309 
1476 
2639 
3800 

4957 
6ni 
7262 
8410 
9515 
0697 
1836 
2972 
4105 



6250 6362 
7374 7486 



8608 
9726 
084 



2627 

3899 
^167 
6432 
7693 
8951 
0204 

H54 
2701 

?944 

5183, 
6419 

7652 
8881 
0106 
1328 
2546 
3762 

4973 
61^82 

7387 
8589 

9787 
0982 
2174 

3362^ 
4548 

5730 
6909 
8084 

9*257 
0426 
1592 

2755 

3915 

5072 

6226 
7377 

9669 

osTI 
1950 
3085 

4218 

5348 
6475 

7599 
8720 
9838 
0953 





►la 


The Logarithms of Natural Numbers to 4400. | 


Nf^ 


1 


I 2 

1176 1287 


3 


4 


5 <5 


7 


8' 


9 




590 S9-»o<55l 


'397 


1510 


1621 1732 


1843 


i9S$ 


2o56 




391 


2177 


22S« 


2399 


t^io 


2621 


27 }2 


2843 


^.954 


3064 


3175 




392 


3286 


3397 


3SO8 


3618 


3729 


^840 


395040^11 


4171 


4282 




393 


4393 


4503 


4613 


4724 


4834 


4945 


5055 


$165 


5276 


5386 




394 


549^ 


"5606 


S717 


5827 


5937 


6047 


6157 


6267 


<5377 


6487 




^^'y 


6S97 


6707 


6817 


6927 


7037 


7146 


7256 


7366 


7476 


7585 




396 


7^9) 


7805 


79»4 


802^ 


8iJ4 


8*43 


8353 


8461 


8572 


8681 




397 


8790 


S900 


9009 


9119 


9128 


9337 


9446 


95 $6 


9665 


9774 




39^ 


9^H3 


9992 


Old 


0210 


0319 


0428 


0537 


0646 


0755 


0864 




399 

400 


600973 


1082 
2169 


1 190 

a277 


1299 

2386 


1408 
2494 


2602 


162$ 
2711 


1734 
2^19 


1843 
1928 


195 1 

3036 




iO60 


* 


401 


3»44 


3152 


3361 


H69 


3577 


3685 


3794 


3902 


401© 


4m8 




4'^2 


4226 


4334 


4442 


4550 


4658 


4766 


4874 


4982 


5089 


5*97 




403 


5305 


5413 


5521 


5618 


573^ 


5843 


5951 


6059 


6166 


6274 




404 


6381 


6489 


6586 


6704 


68 1 1 


6918 


7026 


7133 


7240 


7348 




405 


745 s 


7562 


7677 


7777 


7884 


7991 


8098 


8io$ 


8312 


Hi 9. 




406 


«526 


863} 


8740 


S847 


S9S4 


9060 


9167 


9*74 


9581 


9488 




407 


9S94 


9701 


9808 


9914 


0021 


0128 


0254 


0341 


0447 


0554 




408 


61.06 .0 


0767 


0873 


0979 


1086 


1 192 


1298 


1405 


1511 


16117 




4C9 


1723 


.8z9 


190 


204.2 


2148 


2254 


2360 


2466 


2572 


2678 




410 


2784 


2890 


2996 


3101 


3207 


3313 


3419 


3525 


<;63o 


373^ 




411 


3842 


3947 


40<3 


4159 


4264 


4370 


4475 


4581 


4686 


479^ 




412 


4S97 


')Oo3 


5108 


5213 


5319 


5424 


5529 


5634 


5740 


584$ 




413 


5950 


60s 5 


6160 


6265 


6370 


6475 


6580 


6685 


6790 


689$ 




414 


7000 


7105 


7210 


73^5 


7420 


7524 


7629 
867$ 


7734 


7839 


7943 




41 s 


' 8048 


8153 


8217 


8364 


8466 


8571 


8780 


8884 


89891 




416 


9093 


919^ 


9302 


9406 


9514 


9^15 


9719 


^823 


9928 


0032 




417 


62.0136 


0240 


0544 


0448 


o$5z 


o6$6 


0760 


0864 


0968 


1 072 




418 


1176 


1280 


1384 


1488 


1592 


169$ 


1799 


1903 


2007 


2^1 IQ 




4'9 

420 


2214 


2.318 
33')3 


242; 

3456 


2515 
3559 


2618 
3063 


2732 
3766 


2i?3S 

3869 


2939 
3972 


3041 
4076 


3 If 6 
4179 




3249 




421 


4z8i 


4385 


4488 


4591 


4694 


4798 


4901 


$004 


5107 


5209 




422 


.. ^312 


5415 


«;5i8 


5621 


5724 


5827 


5929 


6034 


6135 


6238 




423 


6540 


6443 


6546 


6648 


6751 


6^U 


695^ 


70s 8 


7161 


7263 




4'4 


7166 


7468 


757' 


7^73 


7775 


7878 


7980 


8082 


8184 


8287 




425 


8389 


8491 


8593 


8695 


8797 


8900 


9002 


9104 


9206 


9308 




426 


9410 


95U 


9613 


97^5 


9817 


9919 


0021 


0123 


0224 


0326 




427 


63.042 >s 


0530 


0631 


0733 


0834 


0936 


1038 


1139 


1241 


1342 




428 


?444 


i54'> 


1647 


1748 


1849 


1951 


2052 


2153 


2255 


2356 




429 
430 


5427 


2558 
3569 


z66o 
367Q 


2761 
3771 


2862 

3872 


2963 
3973 


3064 
4074 


3165 
4175 


3266 

4276 


3367 
4376 




3468 




431 


4477 


457B 


4679 


4779 


4881 


4981 


5681 


$182 


5*83 


5383 




432 


^484 


55^4 
6«;88 


5^^$ 


57^5 


$886 


5986 


6086 


6187 


6287 


6388 




433 


6(588 


6789 


6889 


6989 


7089 


7189 


7189 


73 po 




434 


7490 


7590 


7690 


7790 


88l8 


7990 


8090 


•8190 


8i89 


8389 




435 


8489 


85«9 


8689 


^789 


8988 


9088 


9188 


9287 


.9387 




436 


94?6 


958^ 


968$ 


9785 


9885 


9984 


0084 


0183 


0283 


0382 




437 


<J4.o84i 


0581 


0(J8o 


Q779 


0879 


0978 


1077 


H76 


1276 


137$ 




43B 


1474 


1573 


1572 


1771 


1871 


1970 


2069 


2168 


4267 


2366 


. 


'4^9 


2464 12563] 


26i5? 


2761 h86Q 


2959 


3058I3I56 


3255135541 




•^'^' •^fc •^•^•^» •^^ "^^ •^^ "^^ •^^ •^^ •^^ 




•^» m"^* - "^^ •^^ •^^ "^^ •tf » 



The Logarithms of Natuml Numbers ta 5900. » 1 5 



140 
HI 
542 
545 

544 

545 
54^ 
547 
54S 
549 
5^0 
551 
55.2 
553 
554 
555 
556 

557 

558 

559 

$60 

5^1 
562 

563 
5(^4 
5<J5 
5<56 

5^7 

5(^8 

5.70 

-571 
57i 
573 

5,74 
575 
576 

577 
578 
579 

5« 
581 

582 

58:? 

58< 

586 

589 



731-394 
733-197 

999 
7?4.8op 

735-599 

73<5,99<^ 

7^7.192 

987 

738.781 
739.572 



740:363 
741.152 
939 
742.7*5 
743- 5 lo 

744.293 
745-075 

855 
74^.634 

747-412 



748.188 

74f.73^ 
750.508 
751.279 
752.048 

753.583 
754.348 
755.112 



875 
75<5.<53^ 

757.39<^ 
758/155 

911 
759.^68 
760.422 

761. 17i 

918 

762.679 



763.428 

764. •7<f 

7^5 669 
766. 4n 
767.1 %6 

7<<8.6:?8 
769377 
770.115 



474 
277 
079 
880 
679 
47<5 
272 
067 
860 
651 

442 

230 

018 

804 

588 

371 

153 

933J 

712 

489 



a6^ 
040 
814 
585 
356 

893 
660 

425 
189 

951 
712 

472 
230 
987 
743 
498 
251 
003 
754 



5c»3 
251 
998 

743 
487 
230 
972 
712 
451 
189 



555 
358 

159 

960 

758 
556 
352 
146 
939 
730 
521 

309 
096 
882 
666 

449 
231 

on 

790 
567 



343 
118 
891 
66:^ 

433 

202 

970 
736 
501 
2^ 
027 
7S8 

548 
306 
063 
819 

573 
326 
078 
829 
•568 
326 
072 

8f7 
S62 

304 

046 
786 
5*5 
263 



438 
240 
040 
838 
^35 
431 
225 
018 
810 



599 
388 

»75 
961 

745 

528 

309 
089 

868 
645 

421 

195 
968 
740 
510 

279 
047 

813 
578 
341 



103 
8^4 
624 
382 

'39 
894 

649 
402 

,^•53 
903 



^53. 

400 

U7 

892 

6^6 
378 
120 

860 

599 
336 



7'| 

518 

320 
120 
918 
715 
5n 
305 
097 
1^89 



678 
467 
254 
039 
823 

606 

387 

167 

945 

722 

498 
272 

045 
817 

587 
356 
123 

889 
6$4. 

417 



179 
940 
700 
458 
214 
970 
724 

477 
228 

978 

727 

475 
221 
966 

710 

•;53 

194 

9U 
673 

410 



1^ 
79^ 
598 

400 
200 
998 
795 
590 
384 
177 
96S 

71^7 
545 
332 
118 

902 
684 
465 

345 
023 
800 

57^ 
350 
122 

894 
664 

433 

200 
966 
730 
494 

256 
016 
775 

533 
290 
045 

799 
552 
303 
053 



802 
550 
29-5 
041 
784 

527 
268 

008 

746 

484 



876j9$6 
679 759 



480 

279 
078 

874 
670 
463 

2S6 

047 
836 

624 

411 

196 

980 
762 

543 
323 

loi 

878 

^53 
427 

200 

971 

741 

509 

277 
042 
807 
570 



332 
092 
851 
6p9 
366 
121 

875 
627 
378 
128 

624 
370 
115 
859 
601 

342 
0S2 
820 
557 



560 
359 
157 
954 
749 
542 
355 
126 

915 

703 
489 

274 
058 
840 
621 
401 

179 
955 

731 

504 
277 
048 
818 
586 

353 
119 

883 
646 



408 
168 
927 
685 
441 
196 
950 
702 

453 
203 



.952 
699 

445 
190 
933 
675 
416 
156 

894 
631 



8 

037 

839 
640 

439. 
•237 

033 
828 
622 
414 
a05 

994 
7«2 
568 

355 
1,6 

918 

699 

478 
256 

0J3 

808 
582" 

^>54 
125 

895 

'66'^ 

430 

195 
960 
722 

484 
244 
003 
760 

517 

272 

025 

777 
528 
278 



027 
774 
5^9 
164 
007 
749 
490 

968 
705 



£17 

9«9j; 
720 

519 
317 
^3. 
908 
701 
493 

!!4 

8j5o 

647 
431 
215 

997 
777 
556 
334 
iio 

885 
6>9 

431 
202 
972 
740 
506 
^72 
036 
799 
566 
320 
079 
836 
592 
347 
100 

853 
603 

35*.^ 
loi 

848 

594 
3^3 
082 
823 
S64 

303 
041 

77«i 



f The Logarithms of Natural Numbers to 6900. * 17 


NO. 
64.0 





1 2 1 


383 


45 « 


•i!9 


S»7i655 


_8^ 
722 


.9 
790 


806.180 


248 


3,6 


6ji 


8$ 8 


926 


993 


06, 


129 


'97 


264 332 


400 1 467 


ejz 


807.535 


603 


670 


738 


805 


873 


941 C08 


076 143 


6I3 


808.211 


278 


346 


+ii 


481 


548 


616 633 


75» 


818 


6^4 


886 


953 


02, 


088 


•55 


223 


290! 358 


425 


49» 


5:i 


.809.560 


627 


694 


762 


829 


896 


9631031 


09S 


,65 


810.232 


300 


367 


434 


501 


568 


636 ; 703 


770 


•837 


^:? 


904 


971 
642 


038 


106 


"73 


240 


307 


374 


440 


508 


811.575 


709 


776 


843' 


910 


977 


0^4 


III 


178 


6co 


812.245 


Hi 

980 


22i 
047 


i4S 
114 


ill 
,80 


■HI 


646 

3'4 


7' 3 
38, 


780 
447 


846 
S'4 


913 


6s, 


813.581 


648 


7H 


7.8' 


848 


914 


981 


048 


'i^ 


187 


652 


814.248 


3«4 


381 


447 


514 


580 


647 


7H 


780 


847 


653 


9'3 


98© 


046 


113 


'79 


246 


3,2 


378 


445 


'75 


654 


815.578 


644 


7,0 


777 


&43 


9,0 


976 


042 


109 


655 


816.241 


308 


374 


440 


506 


Sll 


639 


70s 


771 


838 


656 


904 


970 


036 


102 


169 


235 


30, 


367 


433 


499 


657 


817.565 


632 


697 


764 


830 


89b 


962 


028 


C94 


160 


658 


818.226 


292 


358 


424 


490 


556 


622 


688 


754 


8,9 


^M. 


8«S 


610 


017 
675 


083 
74> 


149 
807 


i'5 

'^73 


281 
939 


346 

CO4 


412 
070 


% 


gg? 


8'9.S44 


661 


820.201 


267 


333 


398 


464 


S30 


595 


661 


727 


792 


662 


858 


924 


989 
644 


©S5 


120 


,86 


«Si 


317 


382 


448 


663 


821.5,3 


579 


710 


775 


84, 


906 


972 


037 


,03 


664 


8Z2.168 


233 


299 


364 


430 


495 


560 


626 


C.91 


756 


665 


82^ 


887 


952 


0,7 


083 


,48 


2,3 


279 


344 


409 


665 


823.474 


539 


605 


670 


735 


BOO 


865 


930 


996 


o6i 


667 


824.126 


191 


256 


32, 


386 


4J> 


516 


581 


O46 


711 


668 


„ 776 


84, 


906 


97, 


036 


lOI 


,66 


231 


296 


361 


669 


8*5.426 


491 


^ 


621 


6S6 


Z5i 


8,5 


W6O 


945 


010 


670 


826.075 


,40 


204 


269 


334 


399 


463 


778 


593 


658 


671 


„ 72« 


787 


852 


9,7 


98, 


046 


III 


'75 


240 


305 


672 


8*7.369 


434^ 


498 


563 


628 


692 


757 


821 


.86 


950 


^73 


828.0,5 


080 


,44 


209 


273 


338 


402 


466 


531 


59$ 


674 


660 


724 


789 


853 


918 


982 


046 


III 


«75 


2^0 


675 


829.304 


368 


432 


497 


56, 


625 


690 


7S4 


8i8j88i 


676 


947 


oil 


075 


•39 


204 


26» 


332 


396 


460 .;24. 


677 


830.^589 


653 


717 


>8. 


^ii 


909 


973 


037 


102 


166 


678 


831.230 


294 


358 


422 


486 


55° 


614 


678 


742 


'806 


67? 
680 


870 


934 
573 


998 

637 


c62 

7C0 


,25 

7^4 


ni9 
828 


253 
092 


317 
956 


381 
019 


145 

083 


832.509 


68t 


833.,47 


2,1 


275 


338 


402 


466 


5iO 


593 


657 


721 


682 


784 


848 


912 


975 


039 


,03 


166 


230 293 


357 
993 
627 


683 


834.42, 


484 


548} 61 I 


675 


738 


802 


S66 


929 


^b 


835.056 


,20 


ii!3iZ46 


3,0 


373 


437 


5C0 


564 


^}i 


o^^9i 


7H 


8i7i88i 


944 


C07 


07, 


134 


'97 


261 


68fr 


836.324 


387 


45 J 


5H 


577 


640 


704 


767 


830 


8Qi 


687 


« 957 


020 


0X3 


M6 


209 


273 


336 


399 


462 


f s6 


6B8 


837.588 


652 


715 


77^ 


841 904 


967 


030 


093 


689 


838.2,9 


Z82 


ill 


40S 


47» S34|597 


660 


723 


786 



• c 



•i8 


78* Legaritbms of Natural Numbers to 7400. 




^ 


N°. 





I 


2 


3 


_4, 


_5_ 


li- 


JL\± 


_9_ 


690 


838.849 


91a 


975 


038 


lOI 


164(227 


289 


352 


415 




691 


8J9.478 


541 


5o4 


667 


7*9 


792 


H^ 


918 


981 


043 




692 


840.10^ 


169 


232 


294 


*c^7 


420 


482 


545 


5o8 


671 




693 


733 


795 


850 


921 


984 


046 


109 


172 


234 


297 




694 


841.359 


422 


485 


547 


610 


672 


735 


797 


860 


922 




^95 


984 


047 


no 


172 


235 


297 


360 


422 


484 


547 




696 


842.609 


672 


734 


7p6 


859 


921 


983 


046 
^9 


108 


170 




697 


843.23 J 


295 


3^7 


420 


48* 


544 


tfo6 


731 


793 




698 


855 


918 


980 


042 


104 


166 


229 


291 


353 


41$ 




699 
700 


.844^77 


539 

160 


60 1 
222 


663 
284 


34<S 


788 

408 


850 
470 


912 
532 


22* 
594 


036 
556 




H45.09?* 


701 


718 


780 


842 


904 


956 


028 


090 


151 


2IJ 


275 




70i 


846.337 


399 


4*5 1 


5*3 


584 


645 


708 


770 


832 


»95 




703 


955 


017 


079 


141 


202 


264 


326 


388 


449 


511 




704 


847-573 


534 


696 


758 


819 


88i 


943 


004 


067 


"7 




705 


848.189 


2U 


31* 


374 


435 


497 


559 


620 


682 


743 




706 


805 


856 


928 


989 


051 


II* 


174 


256 


296 


358 




707 


849419 


481 


542 


604 


665 


726 


788 


849 


911 


97a 




708 


850.033 


095 


156 


^'7 


V9 


340 


401 


462 


5*3 


585 




709 


646 


707 


769 


830 


891 


952 


014 


0^5 


135 


197 




710 


851.258 


319 


38« 


442 


503 


564 


6Tj 


686 


747 


808 




7" 


870 


931 


992 


05J 


114 


'75 


23«S 


297 


3^! 


419 


- 


71a 


852.480 


541 


602 


66^ 


7»4 


785 


846 


907 


968 


029 


713 


853.089 


150 


211 


272 


33 J 


394 


455 


516 


576 


6J7 


714 


. ^98 


759 


820 


881 


94* 


002 


063 


124 


ia4 


245 




7'S 


8543o«5 


3«7 


427 


4S8 


549. 


5io 


670 


731 


792 


852 




716 


. 913 


974 


0J4 


095 


156 


216 


277 


337 


398 


459 




7»7 


855.519 


580 


540 


701 


761 


822 


882 


943 


003 


064 




71S 


855.124 


'!^ 


i*'> 


306 


356 


427 


487 


548 


608 


668 




720 


' 729 


789 
393 


850 
453 


910 


970 
574 


OJI 

634 


091 

694 


£5" 

754 


XI2 

815 


*7« 
875 




857.332 




721 


935 


995 


056 


116 


176 


236 


296 


357 


4«7 


477 




722 


858.537 


597 


657 


7l8 


77» 


838 


898 


958 


otS 


079 




723 


859.138 


198 


258 


318 


37« 


418 


499 


559 


619 


679 




724 


739 


798 


858 


9«8 


978 


038 


098 


»58 


218 


278 




72 s 


8^0.338 


398 


458' 


5«8 


578 


tf37 


697 


757 


817 


877 




726 


937 


996 


o$6 


116 


176 


a3<5 


295 


355 


4>5 


475 




727 


851.534 


594 


654 


714 


773 


833 


893 


952 


012 


072 




7l8 


862.131 


191 


251 


310 


370 


430 


489 


549 


608 


668 




729 
7JO 


727 


787 
?82 


846 
442 


9o5 
50T 


966 
561 


025 
6io 


085 
680 


144 
739 


204 
798 


263 
85s 




853.323 




731 917I 


977 


oj6 


096 


155 


214 


274 


333 


392 


452 




732 


864.511 


570 


63« 


589 


748 808 


867 


926 


985 


04s 




733 


85s. 104 


163 


222 


z82 


34» 460 


459 


S18 


578 


637 




73-1 


595 


755 


814 


873 


9J3 992 


051 


no 


169 


228 




735 


266.287 


346 


405 


465 


524 


583 


641 


701 


760 


819 




73« 


878 


?37 


996 


055 


114 


173 


23* 


29, 


350 


409 




737 i 


?67-4<S7 


5s6 


585 


^44 


703 


7.52 


821 


880 


939 


997 




7?8Stf 8.056 1 


"5 


174 


533 


29* 


J 50 


409 


468 


527 58«J 




--0I 


«S,.4'- 


T^■^, 


167 


R21 


870 


528 


007 


o<6 


tia. 


.7, 





740 

741 
742 
743 



The Logarithms of Natural Numbers to 7900. *I9 



869,222 
818 

870.404 
989 

371-573 
1872.156 

/T- 739 
747 873.321 
->iO 902 

8 74-482 

750 875.061 

751 640 

752 875.218 
7§3 795 

754 877.371 

755 947 
7$6 878.522 

757 879096 

758 669 



744 

745' 

746 

747 
748 

749 



759 



880.242 



76d 814 

761 881.385 

762 955 

763 882.524 

764 883.093 

765 661 

766 884.129 

767 795 

768 885.361 

769 9^6 
886.491 



770 ^,- 

771 887-054 

77* ^,J^7 

773 888.179 

774 «„ 741 

775 889.302 

776 ^^'^ 

777 890. 

778 ^ 

780 892. 
781 

, 782 893 
783 

784 894 
785 

786 8f5 

787 - 

788 896. 

789 897 



98c 



651 
.207 
762 
316 
870 
'422 
.975 
.526 
077 



2;;0 

877 
46Z 
047 
631 
215 
797 
378 
960 
540 

119 

698 
276 

853 
429 

004 

579 

153 
716 

199 

871 

442 
012 
581 
150 
718 
285 
852 
4i8 
983 

547 
III 
674 
236 

797 
358 
918 

477 
035 
593 
MO 
707 
262 
817 
371 
9*5 
478 
030 
581 
131 



5^4 
170 

755 

339 
923 

506 
088 
669 
250 

830 

409 
987 
564 
141 

717 
292 
866 
440 
013 

^ 
156 
727 
297 
866 

434 
002 
569 

135 

700 

20'5 

829 
392 

516 

077 
638 

756 

3H 

872 

428 
985 
540 
094 
648 
201 

754 
306 

857 
407 



642 

228 

813 
398 
981 
564 
146 
727 
308 
887 

466 

022 
198 
774 
349 
924 

497 
070 
642 

213 

784 

354 
923 
491 
059 
625 
191 

757 
321 

is] 
448 

oil 

573 

694 

812 

370 
927 

484 
040 

595 
150 
704 

2^7 
809 

361 

912 

462 



8 



701 
287 
872 
456 
040 
622 
204 

785 
366 

945 
524 
102 
680 
256 
832 
407 
981 

555 

127 

69? 

270 
841 
4U 
980 
548 

»>5 
682 
248 
8,3 
3_78 

942 

505 
067 
629 
190 
■750 
309 
,868/ 
'426 
■983 

540 
096 

;65i 

205 

!759 
312 

,864 
416 
967 

517 



760 

345 
930 

515 
098 
6$i 
262 
843 
424 
003 

58I 
160 

737 

889 

4^4 
038 
612 
185 

7je 
328 
898 
468 
036 
605 
172 

739 

305 
870 

.^4 

99§ 
561 
123 

685 
246 
806 

365 
924 
482 
039 

595 
151 
706 
261 

814 
367 
919 

47" 

022 
S72 



•20 Toe Logarithms of Natural Numbers to 8400. j 


N". 





-Ll-iL 


i. 


4 


5 


5 


7 ^ 


J_ 


7;--' 


^97.607 


682 737 


792 


8+7 


902 


957 


012 067 


122 


791 


89^.17^ 


2311235 


341 


396 


451 


505 


55i 5i5 


670 


79* 


725 


780 1 83 5 


890 


944 


999 


054 


109 


•54 


218 


793 


8^9.27? 


328 


?83 


437 


492 


547 


5o2 


555 


7" 


766 


794 


820 


87') 


9>o 


985 


039 


09+ 


149 


203 


V^^ 


312 


795 


900.357 


422 


476 


551 


58^ 


540 


595 


749 


804 


858 


796 


9l^ 


968 


022 


077 


131 


166 


240 


295 


349 


404 


797 


9M.458 


5M 


567 


622 


676 


731 


785 


840 


894 


948. 


79'J|90i.oo3 


057 


112 


166 


229 


275 


329 


384 


438 


492 


799 


547 


601 
144 


<555 

198 


7«0 
253 


7h 
307 


818 
35i 


873 
415 


927 

470 


9ii 
524 


036 
578 


800 


,902.090 


Soil 632 


587 


74 « 


795 


849 


90J 


958 


012 


o5tf 


120 


^21934.174 


228 


283 


1^7 


391 


445 


499 


553 


5o7 


(56 1 


805 


715 


770 


824 


873 


5>32 


986 


040 


094 


M8 


202 


804 


905.256 


310 


j54 


4t8 


472 


S25 


580 


634 


588 


742 


805 


795 


8so 


904 


9$8 


012 


o(5$ 


119 


173 


227 


281 


3o6 


906.33$ 


359 


44? 


497 


ISO 


5o4 


558 


712 


755 


820 


07 


H73 


927 


9ii 


03$ 


089 


142 


195 


250 


304 


358 


^08 


907.41 1 


455 


5«9 


573 


626 


680 


734 


7S7 


841 


895 


809 
810 


94}^ 


002 

539 


056 
592 


109 

646 


163 

69) 


217 
753 


270 

.807 


324 
860 


378 
914 


11} 
967 


908.485 


3ii 


909.021 


074 


128 


lU 


235 


2S8 


342 


395 


449 


502 


812 


555 


609 


6<$3 


7«6 


770 


823 


877 


930 


984 


037 


l'^ 


910.090 


144 


197 


251 


304 


358 


41 1 


464 


518 


571 


S14 


624 


578 


73f 


784 


838 


891 


944 


998 


Oil 


104 


?'^ 


911. 158 


211 


2i54 


?I7 


371 


424 


477 


550 


584 


637 


S16 


690 


743 


797 


850 


903 


955 


039 


053 


Ii5 


169 


817 


912.222 


27$ 


328 


38i 


425 


488 


541 


594 


547 


700 


S18 


7U 


8o5 


8lP 


913 


966 


019 


072 


I2S 


»78 


231 


819 

820 


913.234 


127 

867 


390 
920 


443 

97? 


49«S 
026 


_S_49 
079 


5o2 

131 


^55 
184 


703 
*37 


761 

290 


814 


821 


914-543 
872 


395 


449 


501 


555 


608 


55o 


713 


765 


819 


822 


925 


977 


030 


083 


135 


189 


241 


294 


547 


8.3 


91^.400 


453 


505 


$58 


611 


644 


7i5 


759 


823 


874 


824 


927 


980 


033 


085 


138 


191 


*43 


296 


349 


401 


82$ 


916.454 


507 


559 


612 


664 


717 


770 


82 i 


875 


927 


Si6 


9^0 


033 


0^5 


IJS 


190 


243 


295 


348 


400 


45 3 


827 


917-505 


SS8 


610 


(S63 


7»S 


758 


820 


873 


9^5 


978 


828 


91&.030 


0S3 


135 


i38 


240 


292 


345 


397 


450 


502 


8^9 


5'?4 


607 


660 


712 


764 


816 


869 


92c 


973 


oz5 


S?o 


919.Q78 


130 


183 


235 


287 


340 


392 


444 


496 


549 


831 


601 


653 


70s 


758 


810 


852 


914 


957 


019 


07 » 


832 


920.123 


175 


228 


2S0 


332 


384 


435 


849 


541 


593 


8i? 


645 


691 


749 


801 


853 


906 


958 


010 


062 


114 


'^'?4 


921.165 


318 


270 


322 


374 


425 


478 


530 


582 


534 


83<; 


686 


738 


790 


342 


89+ 


945 


998 


o$o 


102 


154 


835 


922.206 


258 


310 


362 


414 


455 


518 


570 


622 


574 


837 725 


777 


829 


881 


933 


985 


037 088 


C40 


192 


S38 923.244 


296 


348 


399 


451 


S03 


555 <5o7 


658 


710 


S39 162 814I865I 


917 969lo2ilo72li24li76"288| 



191 



m 







_ . . 5^5®^^* ;#! tea :5Sf "~ "' 

T<33n^ii^KHg^^|^^e?| rSia^ LOfiL tfi^^ 
•^fc a^fc «^fc •^k "^^ •^^ "^^ •^^ 



* 22 Hie Logarithm of Natural Numbers to 9400. 



890 

891 
892 

893 
894 
895 

896 

897 
S98 

899 
900 
901 
902 
903 
904 
905 
906 
907 
908 

y>9 



910 
911 
912 
913 
914 

915 
916 

917 
918 

919 

920 
921 
922 
925 
924 
92$ 
925 
927 
928 
929 
930 

931 
932 

953 
93+ 

9^')i 



949.390 

950.3(^5 
851 

951.337 
28g 

9^1.308 
792 

953275 
' 760 



954-242 

725 
955.205 

688 
956.168 

649 
957-128 

507 
958-086 

564 



483 

950 

969.416 

S82 

970.347 
812 

936.971.276 
937! 743 
938 1 972.203 
939 1 665 



I 


2 


3 


4?9 


4t8 


53« 


9i6 


975 


024 


4'? 


463 


$" 


900 


949 


997 


3»6 


43 s 


483 


872 


920 


9<9 


35« 


40$ 


453 


841 


88p 


938 


3»5 


373 


421 


80H 


9$6 


90-; 


391 


339 


387 


77J 


82 1 


869 


aS5 


303 


351 


7?« 


784 


8J2 


2I<J 


2<4 


312 


fij>7 


744 


792 


176 


224 


271 


6?5 


703 


751 


»34 


181 


229 


61 a 


659 


707 


084 


'37 


1S4 


$65 


6.4 


661 


042 


090 


138 


518 


56S 


6'3 


994 


041 


089 


469 


5.6 


563 


943 


990 


033 


417 


464 


5" 


^o 


937 


985 


36? 


410 


457 


8J5 


8S2 


919 


307 


354 


401 


778 


825 


872 


249 


296 


343 


719 


766 


813 


189 


235 


*83 


6si 


705 


7$2 


127 


173 


220 


595 


642 


588 


0(J2 


109 1^6 


^30 


576 523 


996 


043 


090 


462 


509 


556 


929 


975 


02 r 


39J 


443 


485 


85^ 


904 


951 


322 


369 


415 


78^ 


«3a 


879 


249 


295 


342 


712 


758 


804 



4 


5 


6 


7 


8 


58$ 


634 


683 


731 


780 


073 


(21 


170 


219 


257 


5<5o 


608 


<557 


705 


754 


045 


095 


»4? 


192 


240 


532 


580 


629 


tf77 


726 


017 


o56 


114 


153 


211 


502 


550 


599 


547 


696 


985 


034 


083 


131 


180 


470 


518 


555 


61$ 


653 


953 


00 1 


049 


098 


»45 


435 


484 


53* 


580 


528 


918 


966 


014 


o52 


110 


J^5> 


447 


495 


543 


592 


880 


928 


975 


024 


072 


351 


409 


457 


505 


553 


840 


8S8 


93<5 


984 


032 


320 


358 


415 


464 


51' 


799 


847 


894 


942 


990 


277 


32s 


373 


420 


468 


755 


80J 


850 


898 


946 


232 


280 


328 


?75 


423 


709 


757 


804 


^"^l 


900 


1*5 


233 


280 


328 


376 


551 


709 


7J« 


804 


851 


ijd 


184 


231 


»79 


325 


5ii 


558 


706 


753 


801 


08 s 


132 


180 


227 


»75 


559 


606 


<S53 


701 


748 


OJ2 


079 


126 


'74 


321 


504 


5S» 


599 


546 


593 


977 


014 


071 


ti8 


i5< 


44s 


495 


542 


590 


637 


919 


965 


013 


o5o 


108 


390 


457 


484 


«i 


578 


85 


907 


954 


OOI 


048 


329 


376 


423 


470 


517 


798 


845 


892 


939 


986 


257 


314 


351 


408 


454 


735 


782 


829 


875 


922 


203 


249 


295 


343 


,389 


570 


716 


753 


810 


856 


Ij6 


183 


229 


275 


323 


5o2 


549 


69$ 


74» 


788 


oS\i 


114 


161 


207 


254 


533 


579 


625 


572 


719 


997 


044 


090 


137 


183 


4*1 


508 


554 


5oo 


547 


9*5 


971 


oi8 


064 


Ho 


588 


434 


480 


527 


573 


8<i 


«97 


943 


9«9 


1035 



_9_ 
829 
315 
803 
289 

774 
259 
744 
228 

711 
194 

«77 
158 

640 

120 

5oi 

080 

559 
03» 
515 

994 

47' 

947 
4*3 
899 
374 
848 
322 

795 

258 

741 
212 
584 

155 

525 
095 
554 

033 

501 

9*9 
435 

903- 

359 

835 

300 

7*5 

229 
693 

156 
619 

082 



Tbe Logarithms if Natural Numbers to 9900. •a3| 


940 





»74 


' Z 
220 


3 4 
266 3'3 


359 


405 451 


8 

497 


-9. 
543 


973-128 


941 


590 


636 


682 


728 


774 


820 


866 


913 


959 


ooS 


942 


974.051 


097 


•43 


189 


235 


281 


327 


373 


420 


466 


943 


5i» 


558 


604 


650 


696 


742 


788 


834 


880 


926 


944 


972 


018 


064 


IIO 


'A 


202 


248 


294 


340 


386 


945 


975'432 


478 


5*4 


570 


616 


661 


707 


753 


799 


845 


94« 


891 


937 


983 


029 


075 


121 


166 


212 


258 


304 


947 


976.350 


396 


442 


487 


533 


579 


625 


67. 


717 


762 


948 


8c8 


854 


900 


946 


991 


037 


083 


129 


'75 


220 


949 


977-266 


3>2 


3^8 


403 


449 


495 


54« 


586 


632 


678 


950 


724 


769 


81S 


861 


906 


952 


997 


043 


089 


»35 


95' 


978-180 


226 


272 


3>7 


363 


409 


454 


500 


546 


S9« 


952 


<37 


683 


728 


774. 


819 


865 


9" 


956 


002 


047 


953 


979.093 


>38 


184 


230 


275 


321 


366 


412 


457 


503 


954 


548 


594 


639 


685 


730 


776 


821 


867 


912 


958 


955 


980.00} 


049 


094 


140 


i*5 


23 1 


276 


322 


367 


412 


956 


458 


503 


549 


594 


640 


685 


730 


776 


821 


867 


957 


912 


957 


003 


048 


093 


139 


184 


229 


275 


320 


958 


98i.3<J5 


411 


456 


501 


547 


592 


637 


683 


728 


773 


959 
960 


.819 


864 
316 


909 
362 


9H 
407 


000 
452 


04s 
497 


090 
543 


'35 
588 


181 
633 


226 
678 


982.271 


961 


723 


769 


814 


859 


904 


949 


994 


040 


085 


130 


962 


983-175 


220 


265 


310 


356 


401 


446 


49' 


536 


581 


965 


626 


671 


716 


762 


807 


852 


897 


943 


987 


032 


9<54 


984.077 


122 


.67 


212 


257 


302 


347 


392 


437 


482 


9«5 


527 


572 


6,7 


662 


707 


752 


797 


842 


887 


932 


966 


977 


022 


067 


112 


157 


202 


247 


292 


337 


3«2 


96y 


985.42(J 


471 


516 


56. 


606 


651 


696 


74' 


786 


830 


96% 


875 


920 


965 


010 


05s 


100 


'44 


189 


234 


279 


969^ 
970 


98<J.324 


369 
S16 


4>3 
861 


458 
906 


503 
9S» 


548 
995 


593 
040 


637 
085 


682 
130 


727 
1^* 


77' 


P7I 


987.219 


264 


309 


353 


398 


443 


487 


532 


577 


622 


972 


666 711 


756 


800 


845 


890 


934 


979 


024 


068 


973 


988.113 l«S7 


202 


247 


291 


336 


38. 


425 


470 


^A* 


974 


559. 603 


648 


693 


737 


782 


826 


871 


9»5 


960 


975 


989.COS 049 


C94 


138 


• 83 


227 


272 


3.6 


361 


fl 


916 


450 '494 


5?9 


583 


628 


672 


717 


76. 


806 


850 


977 


895 939 


983 


028 


072 


117 


161 


2C6 


250 


294 


978 


950.339 383 


428 


472 


5.6 


56, 


605 


650 


694 


738 


929 
980 


783 


827 
270 


87 J 
3^5 


916 
359 


960 
1°§ 


004 
448 


049 
492 


093 
536 


L37 
580 


182 

6^S 


991.226 


981 


1669 713 


757 


802 


846 


890 


934 


979 


023 


067 


982 


99i.iii 156 


200 


«44 


288 


333 


377 


421 


465 


509 


983 


553598 


642 


686 


730 


774 


818 


863 


907 


951 


984 


995 039 


083 


127 


172 


216 


260 


304 


348 


392 


985 


993.436 480 


524 


568 


612 


657 


701 


H^ 


789 


833 


986 


877 92« 


965 


009 


053 


097 


141 


'85 


229 


273 


987 


994.317 361 


405 


449 


493 


537 


581 


625 


669 


7»3 


988 


. 757 801 


845 


880 


933 


967 


021 


064 


108 


IC2 


9^ 99'i-i96 24ol284'328l372l4«6l46o 


504 


S47>59'| 























•24 


7 he Logarithms e/I^atural Numbirs to loooa | 


N'. 


—I— (.» * 


3 


4 


S 


6 


7 


9 


9 


990 


90^ 03J 1679' 723 


767 


811 


854 


898' 


942 


986 


030 


991 


996.0741117 


161 


205 


249 


293 


336 


380 


if* 


468 


992 


• S>2 555 


599 


643 


687 


730 
168 


774 


818 


^62 


905 


993 


949 


993 


037 


oiio 


124 


212 


«55 


299 


343 


994 


997.386 


430 


474 


S^7 


56. 


605 


648 


692 


736 


779 


995 


?23 


867 


910 


9S4 


998 


041 


o8f 


128 


172 


216 


996 


998.259 


303 


346 


390 


434 


477 


521 


564 


6o« 


652 


997 


69s 


739. 


782 


826 


869 


9<3 


956 000 


043 


087 


998 


999.130 


174 


2l8 


261 


305 


348 


39« 435 478 


522 


P99 ' <6i; 1 


6oQ'6?2'f96| 


739 


2lL 


826 870 913 9571 



A 

T A B L E 



O F 



j^tificial or Logmthmjc 
SINES and TANGENTS, 

To every Degree and Minute of the 
Quadrant, 



*D 



T 'A B L E 



1 



O P 



'Artificial or L^aritbmic 

SI N E S, 

To every Ibegree and Minute of the 
Quadrant. 



Min. 

o 

1 

2 

i 

4 

6" 

7 
8 

9 
lo 

II 

12 

«3 

H 
'5 

i8 
»9 

20 



oDeg. 



6.4637 

6.764^56 

6.940847 

7065786 

7*162696 



7.241877 
7.308824 
7.366816 
7.417968 

7.505^8 
542906 
1 7.577668 
,7.609853 
; 7639316 



TdbU of Lt ^qjHtbmc Stttei 
IS 




2 Dcf . 



83243 

8977s 

96207 

302546 

08794 



7.667844 
7.694173 

I7.718997 
'7-742477 1 
7764754 1 



H954 
21027 

27016 

32924 

38753 



44S04 
50180 

|S783 
66777 I 



8 542819 

4999$ 
S3539 
57C'54 
60540 



65999 
67431 
70836 

66 



5^718800 
21204 

«59?2 
8.730688 



JflJeg. 



f 




9715* 
8-600932 
03489 
06623 
09734 



7667 

„ 9969 
8^742259 



4536 

68ot 

o 9055 
8.751297 

3528 



5747 

8.7601 J 1 
«337 
45" 



8.843584 
5J87 
7Hi 
o»*97l 
8>85Q7$i 

«24 



J*9 
7801 

9546 
8.861 281 



3014 
4738 

8165 
9868 
8.871565 



325 

0615 
8x85 



Min- 



II 
12 
«3 
»4 

16 

>7 
18 

19 
so 



t T A B L E 



r 



Artificial or Logmthmic 

TANGENTS, 

to tvery t>egree and Wnute of the 
Quadrant, 



d ^dU ef Logantbtnic frngHas. 


L 


o 


6 Deg. 


1 Deg. 


»Tkg. 


3 r«8. 

8.7«939« 


4D* 


Mill 
6 


o 


8. f 41921 


i.543083 


8.844^44 


t 


6.463736 


49101 


46691 


«i8o6 


<!4«5 


I 


a 


6.764756 


56165 


50268 


44aQJ 
i6588 


8a6o 


a 


i 


6.J40»47 


6JI15 


51817 


8.850057 


3 


4 


7.065 7«tf 


6995* 


^ 573 J<5 


«8959 


1S46 


4 


-6 


7.^62696 


76691 


60828 


8^31317 


3628 


5 
6 


r-24»«78 


«33«3 


64291 


3^6} 


5403 


I 


7;|o882$ 


89«5« 


677*7 


599« 


7171 


1 


t 


7.366817 


96a9« 


7UJ7 


« *5'7 


8932 


i 


9 


7.4»797o 


8.302633 


745«o 


8.74«3*26 


8.86C686 


s» 


(I 


I'^iT^l 


08884 
.15046 


77877 


Z9t2 


2433 


Id 
II 


7.505 wc 


' 81208 


<ao7 


4172 


tJ 


7.542909 


sun 


84,14 


7479 


, 590$ 


12 


:»3 


7.577671 


27114 


87794 


9740 


7632 


13 


14 


7.6<J9857 


33025 


91051 


8.751989 


- « 9'^ ' 


14 


i6 


7.639820 


38856 


94««i 
« ,9749* 


4227 


8.S71064 

2773 


15 

Tift 


7Myfi49 


44610 


<5453 


«7 


7.694179 


50289 


8,600677 


866ii 


4469 


]l 


i> 


7.719003 


55895 


03839 


8.760872- 


6162 


19 


7.742484 


61430 


06978 


3065 


7849 


19 


2o »7.7«47«i 


66894 


10094 


5»45 


9529 


10 








»Pa"^ 




. ..._... 





L,_. 




J3 
34 

36 

.39 
40 

■r — 
> 41 

-J 4^ 
«iJ"|9#li'-^^J 44 

.«ff .,'##J(5<il 4« 



l:|!iif 



••«•»••«•» m*^*m V^V VqV «-ff^-9*«-W9 

©9 ©Q ^^ ^^ ^^ ©O ©Q 







:|: 



•,«^» 

*";'^, 



Mixu 

21 
22 
23 

28 

29 

31 
32 

BB 
94 

21 

37 
38 
39 

4^ 
42 
43 
44 
45 
46 

47 

48 

49 

JO 

51 

52 
55 
54 

J5 

5^ 
57 
5« 
59 
60 



7. 78708 8. 9^5*5 8. .8340 
7. 95099 8-403338 8-631308 

7.910894 8. 08304 8. 4255 

7. 26134 8. 13213 8. 7185 

7. 408 <> t 8. 18068 8.64CC93 

7. 55100 8. 22869 8l 2982 

7. 68889 8. 17618 8, 5853 

7- 822 < 3 8. 32315 8. 8704 

7. 95219 8. 36962 8.651537 

8*0078 09 8. 4156 8^ 4352 



J Table rf Logaritbmic Tanggn is. 
3l>cg. 



_oDeg. f iD^ 



•25 k, 



7.785951 8.372291 8.613189 

7.806154 «. 77622 8. 6261 
7. 254^ 8- 82889 8. 9313 
7* 43944 ^' 88092 8.622343 
7> 616 74 8. 93234 8. ' 5352 



8. 20044 ^* 46110 8. 7149 

8. 31945 8* 50613 8. 9928 

8* 43527 8. 55070 8662689 

8. 54809 8. 59481 8. , 5433 

8. 65806 8.^849 8. 8 160 

8^76531 8. 68172 8.670870 
8. 8^997 8. 72453 8. 3563 
8. 76693 8. 6!239 



8,107202 8. 80892 8, 8900 



8. 97217 

8,107202 w. ^wwy* »/• wyu^j 

8. 16963 8. 850 50 8.6815-44 

8, 26510 8. 89170 8. 4172 
8* 35851 8, 93250 8. 6784 

8. 44996 8. 97292 8. 9381 

8. 53952 8.501298 8.691963 

8. 6*727 8. 05 267 8, 4529 



8. 71328 8. 09200 8. 7081 

8. 79761 8. 13098 8. 9617 

8 88036 8. 16961 8702139 

8. 96156 8. 20790 8. 4646 

8.204126 8. 24586 8. 7139 



8.204126 

8. 71953 8. 28349 8' 9618 

8. 19641 8. 32080 8. 712083 

8. 27195 8. 35779 8. 4534 

8. 34621 8. 39447 8. 6972 

8. 41921 8. 43084 8. 9396 



2Dcg. 



8.767417 

8. 9578 

8.77>^727 

8. 3866 

8^J995 
8. 8114 
8,780222 
\ 2320 
. 4408 
. 6486 



t. 8554 
^.790613 
8. 2662 
8. 470 1 

_67^ 

8. 8752 
8.800763 
8. 2765 
8- 4758 
8.^7^ 
8. 8717 
8. 0683 
8. 2641 

8- 4589 
8. 65 29 

8. 846 1 
8.820384 
8. 229S 
8. 4205 
8. 610? 



8. 7992 
8^ 9874 
8.831748 
8. 3613 

8> U7 ^ 
^ 732J 
8. 9163 

8.84C99S 
8, 2824 
I9. 4644 



3.881202 

8. 2869 

8. 4530 
8. 61^5 

8- 7835 



8. 9476 
8.891112 
8. 2742 
8. 4366 
^_5984 

8. 7596 
8, 9203 

8.900803 
^. 2393 
8>_3y87 
8. 5570 
8. 7147 
S. 8719 
S.91Q285 
8._ 1846 
8. 3401 

8. 495 » 
8. 6495 
8. 8034 
8. 9567 
8,921096 
8. 2619 
8. 4*36 
8. 5649 
8. 7[56 
8658 
93015$ 
8. 1647 
8. 3134 
8^_ A6ie 

8. 609 i 

8. 7565 

8. ^ 9031 

8. f 40494* 
8. 1952 



Mia 

21 
22 
23 
24 
J5 
26 

27 
28 

29 

JO 

31 
32 

33 

34 

J5 

3^ 
37 
38 
39 
JO 

41 

42 

43 

,44 

46 

47 
48 

49 
50 

5^ 

52 « 

5^ 
54 
J5 
5<5 
57 
58 
$9 



»yi DggfiBmeTSuSs Dyrnt Sf 6, 7> ^» ^ 'Q» i y> la, Ltgaritbm 



5- o 
to 

io 

)o 

40 

_$£ 

6. o 
to 
20 

30 

40 

_!2 

7. *o 
10 

to 
Jo 
40 
_5o 
3. o 
10 
io 
30 

40 
50 

h o 
lo 
10 

30 
40 

50 

0. o 
lo 

JO 

40 

_ |o 

1. o 

10 

30 
40 

2. O 
10 

20 

30 
40 

so' 



8.94.019^ 

95-44W 
96.iH9 
9».1S?3 
W-4497 
p.0 070 40 

01.9235 
C3,io89 
04.^^$ 

05.3&59 
06.4^5 

07'54gQ 
085894 
09.6061 

10.599* 
11.559S 
12.5187 
'3 44 70 

M.35$5 
15.24$! 

|tf.x664 
16.9702 
17.807* 

t8.628 o 

t945}2 
202234 

9992 

21.7609 

22.$09£ 

23.2444 

9^70 
24.6775 
25.37^1 
26.0633 

7394 

^27^094 

28.0599 

7048 

89.3399 

9^55 

30.5819 

Jjji893 

7879 
32.37^0 

9599 
33-5337 



1148 
7«87 
4029 
0276 
6430 
24v^ 

5^473 
4366 
0176 
S906 
34.0996 155^ 
^579j7>34 



1738 

5894 
9600 
2883 
5768 
8*78 



0435 
:I256 

3762 
49^ 
5»85 
6^33 



69Z2 

70165 
6973 
6656 
6I25 
5387 



4453 
3330 
2025 
0546 
8900 
7392 

5^^ 
3017 

0759 

8363 

58^3 
3<72 



0386 

7478 

4453 
1314 

8065 
4708 



3x74 
7*84 
0947 
4189 

703^ 
95JO 

1^92 

3421 

4«95 
6071 

69^2 

7583 



7947 
8066 

7951 
76ii 
7060 

6303 

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6 

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5124 

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Logarithmic Tm^0Mts Dggrus^ 5^ 6, 7, 8» 9, ic, : 



5. o 
10 

90 
30 
40 
50 



6. o 

10 
10 
30 

40 
SO 



7. o 
10 
20 

30 

40 

—12 
S, o 

10 
20 
30 
40 
50 

P- o 
io 
20 
30 
40 

_5o 

!©• O 
10 
20 
30 
40 
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II* o 

10 
ao 
30 
40 
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12. O 

10 
20 
30 
40 
$0 



9^.6267 

9^3S77 

^.00.9298 



OS. 1620 

04.5284 
€5.6^59 
06,7752 

07.«57^ 



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11.9429 
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1 8.3059 
1 9' 1462 



9712 
20:7816 

xi-5779 

22.3606 

23.1902 

8871 



24.6319 

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26.0862 

7967 
974964 
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8652 

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30.1951, 

84^3 
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32.1222 



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0529 
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9326 
6Q13 
2607 
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55^3 
1852 

8095 
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*3^ Logarithm Sines^ Peg. 13, 14, 15, 16, 17, 18, 19^ 20. Index 9- 




^ t^ar. ^ines^T^. 21, 2X, a 3, 2^ 15^ zbj 27^ a«/ A. 9? ' 

i~^ ^ 'I'M A|.j_-l| ^ ^ ig^ 



II. o 

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24.0 
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; 9^4 2 
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»/i Day ti PubKJbed^ 

Neatly printed, in 4 Vols. 12m. Price bound in Cal^. u t. 

EmbeUiihed with near lOo Cdpperuplttei, curioufly engraTel 
by the bdk Mailers. 

Tranflatcd from the t^tnchf by John Kelly, Efqj of the ImuP" 
Temple \ Dr Bellanrf of St John's College^ Oiffirdi and J. 
Sparrow, Surgeon anci Mathematician. 

NATURE DELINEATED : Being Philofophkal 
Convcrfations, wherein the wonderful Works of ProvU 
dence, in the Animal, Vegetable and Mineral Creation are hid 
open, the Solar and PlaneUiy Syftcm, and whatever is curious 
in the Methematicks explained. The Whok bcii^ a Compleat 
Courfe of Natural and Experimental Philofophy, calculated for 
the Inftruflion of Youth, in order to prepare them for an early 
Knowledge of Natural Hiftory, and create in their Minds an cx- 
akcd Idea of the Wifdom of the G » E at C r b Xt o »• 
Written by way of Dialogue to tender the Conception more 
familiar and eafy. With a partkularTaUe of Contents tacftch 
Volume, 
Imulon: Printed for J. Hodget, at AtLnihg-GUfi on landon^Bridge^ 

Where tikewife may be had, Juji Tuhlijhed^ 

Dedieatid andprefentei to the Hon, Society of the Imn ftmpU^ 
(Price bound Two Shillings and Six Pence.) 

A New Treatife of Husbandry, Gardening, and other cu« 
rious Matters relating to Country Airain : Containing^ 
A Plain and Pradical Meth^ of Improving all Sort of Meadow^ 
Pafture, and Arable Land, C^r. And making them produce 
greater Crops of all Kinds, and at much lefs than the prefent£x« 
pence. Under the following Heads : I. Of Wheat, Rye, Oats^ 
Barley* Peafe, Beans, and all other Sorts bf Grain. Il^ Tur* 
nips, (Jarrots, Buckwheat, Clover, Hemp, : Rape, Flax and 
Colefeed, i^c. \IV Weld ox Would, Woad or Wade, Mad- 
der Saffron, c^c. IV. Meadow, Pafture Grounds, and the dif- 
ferent Manner of Feeding Cattle and making other Improve- 
ments agreeable to the Soil of the feveral Counties in Great 
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ral, and Creenhoufe Plants. VII. A Curious Scheme of a Farm, 
the Annual Expenceof it, and its Produce. With manyNewt 
Ufeful, and Curious Improvements, never before publiflied. 
The Whole founded upon many Years Experience. By Samuel 
trowel, Gent. To which are added. Several Letters to Mr 
Thomas Liveings, concerning his Compound Manure for Land^ 
with (bme Pra^calObfervations thereon; 



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21 


'9 


18 


17 


t6 


15 


»3 


35«o 


50 

60.0 


12 


II 


to 


J 


7 


6 


5 


4 


2 


t 


3590 


ol 








■~\-\ 








3600 



Hours 


D««.» 


M. 


Deg. 'hM.) 


Deg.' 


I 


2 30 


I 


3 


3« 


I 18 


u 


5 oo 


2 


5 


3a 


I 20 


III 


7 3° 


3 


8 


33 


I 23 


IV 


10 oo 


4 


0^ 10 


34 


I 25: 


V 


12 30 


5 


13 


35 


I 28 


VI 


IS oo 


6 


1,5 


36 


I 30 


VII 


>7 3° 


■7 


18 


37 


1 33. 


VIII 


20 oo 


8 


20 


38 


1 35' 


IX 


22 30 


9 


23 


39 


I 38 


X 


25 CO 


10 


25 


40 


I , 40 


XI 


*7 30 


II 


• 28 


4" 


I 43 


XII 


30 00 


IS 


30 


42 


I 45 


XIII 


3« 30 


•3 


33 


43 


I 48 


XIV 


35 00 


>4 


35 


44 


I SO 


XVi ' 


37 30 


J5 


, 38 


45 


1 53 


XVEl 

xvil- 


40 00 


16 


c 40 


46 


i 55 


4» 30 


»7 


43 


47 


« 58 


XVIII 


45 0® 


18 


45 


48 


2 00 


XIX 


47 30 


•9 


48 


49 


a 3 


XX 


50 00 


20 


50 


SO 


2 5 


XXI 


s* 30 


21 


^3 


51 


2 s> 


XXII 


55 00 


22 


55 


52 


2 10 


XXIII 


57 30 


23 


5« 


53 


2 13 


XXIV 


60 00 


24 


I 00 


54 


2 15 






25 


« 3 


55 


* l» 






26 


I 5 


56 


2 2C 






27 


I » 


57 


2 ^(3 






18 


1 10 


58 


2 2^ 






29 


* 13 


59 


Z 2fe 




- 


30 


1 i$l6o 


2 30 



/H